Ukraine Turmoil Essay

Effects of Weather on Mood

Upgraded College Writing Cause and Effect Essay September 27, 2012 Effects of Weather on your Mood Many of us have mind-sets that are alterable like the tides, and like the tides, for a large number of us those states of mind are directed by numerous things including the climate. While you may feel that you ‘just woke up on an inappropriate side of bed' it might in certainty be that there are different things having an effect on everything here †diet, tiredness, climate and the sky is the limit from there. Actually you could most likely venture to state that the side of bed that you escaped is probably not going to truly have had a lot to do with your disposition at all.Weather is one especially enormous marker of how your mind-set is going to show up for the afternoon. The first and most notable manner by which climate influences mind-set is in what is known as occasional full of feeling issue or SAD. This condition can likewise be known as winter sorrow, winter blues or regular melancholy and fundamentally it depicts a condition wherein the individual finds their mind-set so attached to the changing of the seasons that they in certainty display side effects near gloom each winter.There are different clarifications with respect to why somebody may encounter SAD, however by and large it is accepted to identify with the measure of light. This at that point implies that it isn't in actuality the cold of winter that causes the downturn, but instead the absence of light getting into your mind. Simultaneously temperature can likewise influence state of mind and this is because of vitality use. In the winter our invulnerable frameworks are going to invest more energy so as to keep our bodies warm and our pulse will speed up.All this implies vitality is coordinated toward those undertakings as isn't accessible in as huge amounts for different exercises. Simultaneously this is additionally why you are bound to turn out to be sick your insusceptible framework is as of now under tension from the cold and therefore it turns out to be less ready to fight off the assaults from microscopic organisms and infections. So as to abstain from getting sick at that point and to evade the downturn that can emerge out of it, try to eat heaps of nutrients and minerals that can assist with boosting your safe framework and to get a lot of sleep.At a similar time make a point to utilize bunches of warming so as to warm the house and to keep ailments under control. Rest is likewise influenced by the climate and this thus can cause us bound to feel discouraged. In the event that it is cold for example, at that point you are bound to rest all the more daintily and that gives your body and psyche less quality time wherein to recoup from your day's exercises. Again you are bound to endure ailments and to have low vitality, yet this can likewise bring about cerebral pains and awful moods.Furthermore you will get up in obscurity when your body is instructing you to return to rest thus hormonally you are caught off guard for the afternoon. There are innumerable various manners by which the climate can influence disposition at that point and this remembers direct impacts for state of mind and hormones, just as progressively unobtrusive second request impacts. Ensure that you remain warm and evaporate and that you make for lower vitality in your eating routine.

Important Tips To Write My College Term Paper

Important Tips To Write My College Term PaperYou've got plenty of time and plenty of money to spend, but if you really want to write my college term paper then you must read this first. There are a few simple things you must do before you can start writing your term paper.A mistake many people make is to try and wing it on their papers without really checking what grades they are currently earning in their class. If you really want to get the most out of your college experience then you should spend the first few weeks of term writing your term paper paying close attention to your grades in school.Every term has a lot of interesting aspects to it that you may not be able to see on the list of assignments. Make sure you spend some time researching the options you have for studying, and try to find out what the recommended method of study is for your class.When you get to the end of the year and are considering which courses you want to study for next term, you will find that your work load will be more intense than it was the previous term. It is possible to add extra study time to your load, but make sure you only spend time on the courses you really want to go through. If you don't do this then you will end up wasting time on courses you don't need to be studying.When you get to the end of the term you may find that your paper has been accepted by your university. Congratulations, and you now have your first term paper of the semester written! Now you need to do some more research to help you understand the details of your assignment.Students are often nervous about making any kind of big decision for themselves. Writing a term paper is a major decision, and it is no different. Make sure you understand what is expected of you when you are working on your paper, anddon't rush into things.Good research is a must, and you will find it hard to get all of the information you need on your paper without having done some research on your own. Keep the ideas you have f or topics, and the sources of information for those ideas, together with a number of references you can draw from. The idea behind this is that you can be much more efficient in compiling the facts on your paper.By following these tips you will be sure to write your term paper successfully. The hard part will be the last section, which is how to complete the project.

Perceived Effectiveness Of Influence Tactics Of The United...

Perceived Effectiveness of Influence Tactics in the United States and China is a quantitative research study written by Ping Ping Fu from Chinese University of Hong Kong, and Gary Yukl from State University of New York at Albany. This research study is published in a book named The Leadership Quarterly. INTRODUCTION Ping Ping Fu and Gary Yukl believed that people from different cultures deal with the difficulties of exercising influence differently. According to Smith and Peterson, to understand cultural differences, international relationship, and various influences in cross-cultural cooperation are necessary for managers under 21st century globalization. The authors also believed that the managers who have strong cultural awareness†¦show more content†¦The dependent variables are the tactics effectiveness for a variety of contexts, includes the tactics include a variety of relevant tactics, such as rational persuasion, exchange, coalition, upward appeals, ingratiation, pressure, consultation, inspirational appeals, and personal appeals, because the tactics effectiveness in these contexts is influenced by the manager’s nationality. This research paper does not include any hypotheses; instead the authors explored seven research questions. 1. Are some tactics considered more eff ective by Chinese managers than by American managers? 2. Are some tactics considered less effective by Chinese managers than by American managers? 3. Are the tactics considered most effective by Chinese managers than by American managers? 4. Are there any culturally specific influence tactics used by Chinese managers but not by American managers? 5. How strong is the effect of national culture on perception of influence tactics (in relation other situational and individual determinants/)? 6. How well can the nationality of a manager be predicted from his or her pattern of ratings on tactic effectiveness? 7. How useful are fixed-response scenarios for studying cross-cultural differences? The reason why the authors use the research questions instead of the hypothesis is because research questions helps to explain the purpose of the research. Additionally, this research is a descriptive study, which means, the

We Must Put More Human, Material And Electronic Resources...

Discuss the proposition â€Å"If we could put more human, material and electronic resources into intelligence the more problem of terrorism would disappear. Enhancing Intelligence Management, Developing Community Resilience FAHD PAHDEPIE Terrorism is an evolving and multifaceted phenomenon (Lentini, 2003). Although there is no single definition that is received full approval from academic and governmental circle, most scholars and practitioners believe that the key idea of terrorism is a politically motivated violence against non-combatants that is designed to trigger fear and anxiety among them (Lentini, 2013; Schmid Graaf, 1982). Bakker and Veldhuis (2012) argue that terrorists do not utilize violence to kill or wound their†¦show more content†¦The term of ‘fear management’ in counterterrorism debates is related to the concept of ‘community resilience’ (Bakker Veldhuis, 2012). Borrowing from the concept of ‘resilience’ in ecology, engineering, physics and psychology, where the term is already well developed, the phrase of ‘community resilience’ can be defined as the demonstrated capacity for a given system, such as community, to withstand and respond po sitively to fear and anxiety (Wickes, Zahnow, Mazerolle, 2010). In the counterterrorism context, community resilience can be presumed as a positive factor that limits the negative impact of terrorism on individual and society (Bakker Veldhuis, 2012). In linking with the intelligence as one of the key elements in counterterrorism efforts, there is a big question for the intelligence community regarding what portion that they can contribute in developing the so-called resilient community? While realizing that the need to develop better intelligence management—to put more human, technological, and financial capitals—is immediate necessity (Ackerman, 2001), intelligence community should aware that the major objective of intelligence management is not about apprehending the state of terrorism through data mining and covert actions, rather to develop preparedness, vulnerability, and recovery of the society to face and handle the threats and disorders (Scott Jackson, 2004). Consequently, intelligence should be perceived as an

Malnutrition in Children free essay sample

In 1994 Kevin Carter, a journalist posed a photo in his mail â€Å"Pulitzer prize† wining photo taken in Southern Sudan. The picture depicts of famine stricken child crawling towards a United Nation food camp located a kilometer away, the vulture is waiting for the child to die so that it can eat it. This picture shocked the whole world. No one knows what happened to the child including the photographer Kevin Carter who left the place as soon as the photograph was taken. Three months later he committed suicide due to depression. With this example, poverty is a major world problem. People or families of low income household food it is not enough. Children especially under five year old are the most vulnerable to malnutrition because of their high demand of energy and protein. Malnutrition still remains at alarming rate in the childhood in Mathare slums despite the fact that free food and free medical services are offered and this is the reason why the researcher is much concerned with the need to investigate the causes and impact of this condition in children under five years in this area i. We will write a custom essay sample on Malnutrition in Children or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page . Mathare slums. The researcher at the f this study will aim at making a way forward to the policy makers on how services delivery will be improved and how these malnourished children be handled and rehabilitated. 1. 2 Statement of the problem Causes and impact of malnutrition in under five year old children continues to be the major health problem in the world and in the nation (Kenya) particularly in Mathare slums and other similar areas which is majorly composed of low income earners depicting poverty. The government of Kenya (GOK), in concurrence with other non governmental organizations (NGOs) has donated food, taught people on how to achieve a balanced diet regardless of the low income, and improved nutrition education through antenatal clinics to ensure that children are well nourished. Despite tall these efforts, a survey through Mathare slums and its neighborhood reveals a high number of cases of malnutrition. It is therefore, with this in mind that the study seeks to find out the causes of malnutrition so that the health of children can be restored. 1. Purpose of the study and objectives of the study The main purpose of the study is to find out the causes of malnutrition of children under five years old in Mathare slums. 1. 3. 1 Objectives of the study i. To find out the socio- economic status of the parents of malnourished children ii. To find out how knowledgeable the parents are about nutrition status of their children. iii. To investigate the sources of information for t he parents of malnourished children iv. To identify the steps taken by the parents to curb malnutrition in Mathare slums 1. 4 Significance of the study Many health centers do not diagnose or detect the type of malnutrition from moderate malnourish and severe malnourish. This study will there fore investigate the causes and impact and the commonest type of malnutrition affecting under five year old children in Mathare slums. The results from this study will be used to benefit the health workers and the parents of children below five years old in Mathare slums in the following ways: i. Helps the mothers or parents of these children gain positive insight, that is such condition need immediate medical advice to be solved should it occur because of its short and long terms impacts ii. Helps the health workers and the community to actively respond and attend to the rising condition in below five year old children and young affected. iii. To provide a platform for the public to understand that early treatment or management of this condition will minimize the complication. iv. Helps the researcher establish the uniqueness of this condition in the way it presents, how it is managed and how the patients should be rehabilitated in case of severe complications of malnutrition set in. 1. 5 Scope and delimitation There are several causes of malnutrition but the research will only concentrate on the socio economic factors, knowledge of parents on nutrition, the sources of information and the methods employed by the parents to curb malnutrition. The research will also be confined to children below five years old and their parents especially mothers who will be found in Mathare slums health center. The study could have been extended to below 12 years old as they are medically classified as pediatrics (children) but the researcher’s concern will be on children under five years old because they are the most vulnerable groups among the other children. Mathare slums have other smaller communities but not all of it will be covered because the residents of Mathare leave under similar conditions. The findings can then be generalized in the whole area. 1. 6 Limitation of the study The study is likely to experience many challenges. This is due to the nature of life styles in Mathare. The area is densely populated with many cases of criminal activities. This might make it insecure for the researchers. The members will also be suspicious of the researcher who is a stranger in the area so may not volunteer information as required. Since this is an informal settlement area is congested. This makes the movement very difficult. Sanitation is another major issue. Both solid and liquid wastes are not properly managed. Raw sewage is flowing in open drains. There are no proper latrines for proper excreta disposal, making the whole place full of stench and unsightly scenes. The community members might need money for them to give information. This might be a challenge since there is no enough money for the research. The muddy environment where the research will be carried out makes moment difficult. 1. 7 Assumption of the study i. The information from the respondents will be assumed to be accurate and true to the best of their knowledge. ii. It will be assumed that the communities especially the mothers will be willing and allowed their children for the nutritional assessment. iii. It will be assumed that the respondents now know the causes, impact and prevention of malnutrition. iv. It will be assumed that the mothers of children under five years old will be able to detect early malnutrition. v. It will be assumed that health workers will offer maximum support to the under five years old children and their parents. 1. 8 Conceptual framework Figure 1 * Socio economic status of parents * Knowledge of parents on nutrition Causes of malnutrition among children under five years old in Mathare slums * Cultural practices and attitudes attitude towards certain foodstuffs Malnutrition * Sources of information on nutrition * Methods employed to curb malnutrition 1. 9 Definition of terms Malnutrition: this is a state resulting from intake of nutrients. Children/child:an infant from 0-5 years old. Impact:Is the result of particular influence Mortalityis the number of death in a given period. Assessment:is the process of evaluating nutritional status of an individual. Anorexia:it’s a state of having poor appetite. Health:is a state of complete physical, mental and social well being of a person but not merely absence of disease or infirmity. Prevalence:is the number of affected persons present in a population at a specific time. Pre-disposing:is to make susceptibility to an infection. Anthropometryis the measure of body physical dimension. Protein, energy Malnutrition:This is the condition that comprises of malnutrition that occurs due to imbalance between protein and energy. Nutritionis a study of sum process involving taking of food.

Is The World Running Out Of Oil

Introduction Scientists, geologists and other opinions makers have for years been trying to comprehensively answer the question on whether the world is running out of oil. To this day, there is no comprehensive answer to this question. This research paper argues that the world will not run out of oil as predicted by some environmentalists and geologists.Advertising We will write a custom research paper sample on Is The World Running Out Of Oil? specifically for you for only $16.05 $11/page Learn More This argument is based on the fact that although the world has had its fare share of prediction that oil will be depleted sooner, the discovery of new oil fields, and the improvement of new technology which allows for more oil exploration and exploitation have proved that the world still has enough oil resources. Research method This research is based on the review of existing literature Literature review The 20th century was awash with predictions that sugg ested that the world was facing an imminent shortage of oil. Most of these predictions have already been proven as false especially as more oil has been produced in the recent years triggered by more discoveries of oil fields. Deming (b) however observes that even though the predictions have been proven wrong so far, there are new predictions that cast a lot of doubts about the longevity of the oil resource. Key among them is the Hubbert model, â€Å"which assumes that like all natural resources, oil is limited and finite† (1). Deming (b) does not agree with this model. He argues that while conventional oil reserves may be finite, there is no precise way of measuring the ultimate amount of the resource making it even harder to measure it against the world’s consumption of the same. He also points out to the fact that the cumulative production of oil in the last 50 years has been outdone by the sizes of crude oil resources. This has led to an increase of oil reserves in the world. Another notable author who disagrees with the Hubert Model is Bradley, who states that the theory is disapproved by the functional theory (1). But what exactly does the Hubert model state? Well, according to Bradley, the model is bell shaped and seeks to represent the trend that oil production will take over the years. In the model, Hubert predicted that the 1970s would experience a peak in oil production.Advertising Looking for research paper on business economics? Let's see if we can help you! Get your first paper with 15% OFF Learn More This was a right prediction. He also predicted that the US would experience a peak in gas production in the 1970s, but this has long been proven as an errant prediction. Another of his predictions that oil production would reduce considerable beginning in 2000; this has not been proven yet (Bradley 1). The functional theory was developed by Erich Zimmerman and states that any static interpretation of a natural resource i s futile because resources change with social objectives, react to revised standards of living, alter depending with the expertise or knowledge of the people handling them and change as new technology or arts are discovered. In this theory, Zimmerman suggested that man created resources through his knowledge and hence as his knowledge level increase, so would be the resources (Bradley (a) 6). This theory is supported by Deming (a), who argues that the reason why the world has not yet run out of oil despite the many predictions suggesting otherwise could lie in the new technological development. He also states that high oil prices always signal an impeding shortage of supply thus sending oil explorers back to the fields (1). Among the notable oil scares made in the 20th century include the 1916 Model-T scare, the 1918 Gasless-Sunday scare, the 1920-1923 John-Bull Scare, the 1943-1944 Ickes-Petroleum reserve scare, the 1946-1947 Cold-War scare and the 1947 Winter scare. In the 1970s, the world faced similar scares caused directly by politics in the Middle East. They include the 1973 Arab-embargo oil scare, and the 1979 Iranian-revolution oil scare (Deming (a) 2). The ‘predictions of doom’ on oil production have not had an entirely bad effect in the world. Deming (a) notes that the predictions have led to more innovations and today, men know that they do not only have to rely on conventional oil resources but can as well rely on unconventional resources such as oil shales and tar sands for production of oil. Tar sands The exploitation of tar sands began in Canada in 1967 (Deming (b) 4). Over the years, this unconventional oil resource has proven that quite huge amounts to last the world’s oil needs for the next 1000 years can be produced from the same. Although it was challenging at first usually pushing the cost of production high, the newer technology employed currently has brought the costs down while increasing the amount and rate of produ ction. According to Lomborg, the extraction of oil from tar sands in Canada has helped in pushing the prices of the commodity down from 28 US dollars a barrel in 1978 to 11 US dollars a barrel in 2000 (128). Oil shale Oil shale may be more expensive and difficult to extract, but the entire resource in the United States alone is estimated to provide for the country’s oil needs for a period of 26,667 years (Deming (a) 5).Advertising We will write a custom research paper sample on Is The World Running Out Of Oil? specifically for you for only $16.05 $11/page Learn More Citing information from the Energy information Agency in the US, Lomborg states that current estimates show that a combination of shale oil and tar sands can help in the production of 550 billion barrels of oil, with the production cost estimated to be approximately 30 US dollars (128). This would then lead to a 50 percent increase in global oil reserves. Lomborg further reveals that the total size of recoverable shale oil through out the world is unbelievable to most people who hold the opinion that the world will run out of oil (128). In the entire world, it is estimated that there are 242 times more resources in shale oil than contemporary resources, which could last the world 5,000 years or more. Just like Deming (a), Sah makes a distinction between oil reserves and oil resources and states that those who predict that the world is running out of oil do so by judging prevailing world consumption of oil against the available oil as available in the reserves (242). By doing this however, they miss the mark because resources are more wide and probably a lot of oil resources are yet to be discovered. The fact that exploration for oil remains an ongoing process means that no geologist or environmentalist can correctly predict the extent of the resource. According to Sah, â€Å"reserves are petroleum (crude and condensate) recoverable from known reservoirs under p revailing economics and technology† (242). The reserves can be equated to any past reserves plus additions to reserves minus any production from the same reserves. Accordingly, oil reserves can change as new technologies for oil exploitation keep on being discovered. The reserves can also be added through the discovery of new oil fields; discovery of new reservoirs; the extension of reservoirs in existing fields; and redefining reserves using newer oil extraction technologies. Tertzakian is another writer who believes that though the world is no where near running out of oil, it is the world unquenchable thirst for ‘cheap oil’ that places much pressure on explorers (8). According to him, the world is better off agreeing that oil in future will take more to find, to drill and exploit. As such, he says that price spikes and other incentives can be used to encourage oil companies to find and drill even more oil wells. Like Lomborg notes, man becomes better in exploit ing resources once he is pushed by the need to do so (125). As such, the world can expect that newer and more efficient technology will keep on being discovered for the sake of finding better ways of not only exploring oil, but exploiting already existing oil fields. Technology is especially significant in the exploration of oil in fields that had been deemed too complex to drill or where the oil drilling process was deemed too expensive.Advertising Looking for research paper on business economics? Let's see if we can help you! Get your first paper with 15% OFF Learn More Analysis and conclusion As observed in the literature review section and unlike what some environmentalists and geologists would make the world believe, there is no immediate need to worry about running out of oil because as Lomborg puts it, â€Å"judging what is left of the oil resources is akin to looking at someone’s refrigerator and arguing that once the food therein is finished, then they will starve to death† (125). To justify the thesis of this research paper, it is important to observe that the oil resource is not a finite entity because in addition to already discovered oil fields, there may be numerous others lying undetected and it can only take exploration and maybe some newer technology to discover them. Since exploration will only be spurred by the need for extra production beyond what the world already has in the reserves, it will take a reduction in production for explorers to spend the money that usually go into the exploration process. Until explorers declare that no more oil fields are being discovered, it is immature for geologists or any other person to declare that indeed the world is running out of oil. Based on the arguments of the authors analyzed herein, it is clear that even those claiming that the oil resource will soon run out have no solid evidence to base their claims. More to this, is the fact that none of the predictions some made as early as 1850s have never come true is evidence enough that after all, no one has the abilities to foretell when a natural resource like oil cannot be produced anymore. This is because as oil exploration continues, the chances of newer oil discoveries increases. New technology development also gives us the assurance that just as long as innovators continue searching for better ways to explore and exploit oil; the world could very well be able to go back to the oil wells and reserves to exploit more oil using more efficient technology. Annotated Bibliography This section gives a brief overview of the sources used in the essay. It contains seven sources (books and journals) which have addressed the oil resources agenda extensively. In this section I have identified the author(s), his argument and his conclusion. Most of these sources have considered the views of oil doomsters who claim that oil is a finite resource and based on this, I have come with proven evidence that shows otherwise. Bradley, Robert. â€Å"Are we Running Out of Oil?† Property and Environment research Center Reports September (2004): 3-6. This report is written by researcher Robert Bradley in response to a National Geographic published in June 2004 claiming that the world was witnessing an end to low-cost oil. Bradley claims that though environmentalist and geologists have always published alarming reports about the extinction of oil, none of their predictions have come true so far. He gives the example of geologist King Hubert who had predicted that oil production would be at its peak i n the 1970, but would hit a deep decline in 2000. According to Bradley, the geologist’s predictions have already been proven wrong by oil production in recent times, with 2003 recording a 2.5 production than was the case in 2000. Supporting the argument that the world is not indeed running out of oil, Bradley cites the work of different authors chief among them Erich Zimmerman. According to Zimmerman’s work, people should not assume that the oil resource is fixed since the extent of natural resources is not known to man. â€Å"Resources are highly dynamic functional concepts; they are not, they become, they evolve out of triune interaction of nature, man and culture in which nature sets outer limits†¦Ã¢â‚¬  (Zimmermann 814-15, cited by Bradley 4). In conclusion, Bradley argues that even the lowest oil production periods in the world were not a result of the scarcity of the resource, but as a result of government interventions that blocked oil production hence di storting the oil market processes. Deming, David (a). â€Å"Are we Running out of Oil?† Policy Backgrounder 159 (January 2003): 1-14. Writing for the National Center for Policy Analysis, Deming starts his article by acknowledging that oil resources are non-renewable. He further observes that for more than 150 years, â€Å"harbingers of doom† who include geologists and environmentalists, have been predicting that the world’s oil resources would run dry sooner or later. He however states facts that contradict the positions posed by the scientists. Key among these facts is the continued production of oil throughout the world especially as the 20th century came to an end. Deming also points out that by adjusting oil prices to inflation, it is clear that the prices for petroleum products, gasoline and other oil related products are lower now than they were 150 years ago. His third fact supporting the analogy that the world is not running out of oil is the fact that t he total oil endowment in the world has witnessed a significant increase. As a result, the discovery of new oil resources is higher than the amount that oil driller can take from the ground. To Deming, the ‘gospel’ about extinction of oil has become a favorite past time among prognostics. He for example observes that the US population had been issued with an oil shortage warning even before the first well was ever drilled in the country back in 1859. The writer observes that nature has always proved people who predict an oil shortage wrong through events like the oil glut that was observed in the world in the 1990s. Deming, David (b). Oil: Are we running out? Second Wallace E. Pratt Memorial Conference â€Å"petroleum provinces of the 21st century. Jan. 2000. Web. Deming has written about oil extensively. He specifically appears incensed by people who keep claiming that the oil resource is finite without clearly understanding their allegations. According to him, oil ju st like other fossil fuels can be categorized as either reserves or resources. He defines reserves as the identified oil resources, which awaits extraction and exploitation. Accordingly, he states that reserves keep on expanding as new technological innovations are made in the world. Resources on the other hand include the oil field which have already been identified and those that are yet to be identified. It is because of the definition that he gives to the former that he takes offence with people who keep predicting a looming oil shortage or extinction based or already identified reserves. Citing research from other writers, Deming has done a good job of proving that crude oil reserves in the United States alone grew at the same pace as the consumption rate from 1915 to 1995. Like other authors who oppose the notion that oil is a finite resource, Deming observes that the 20th century was full of ‘false’ predictions which have been proven wrong the cumulative producti on of oil in the same period. He also observes that even if conventional sources would be depleted, man has proven that he can successfully use unconventional resources such as tar sand to produce oil. The author does not doubt that there could be interruptions in supplies and prices as the world tries to make the transition to unconventional sources of oil, but he does state that the world can be sure that the oil potential in the unconventional sources can support the petroleum needs in the world for up to one thousand years. Lomborg, Bjorn. The Skeptical environmentalist: Measuring the real state of the world. Cambridge: Cambridge University Press, 2001. Print. In the ‘optimists and pessimists arguing’ subsection of his book, Lomborg addresses the question of why people are bombarded with news about running out of oil, while it actually never happens. He notes that it is odd that human kind is using more and more oil, yet available statistics show that there could be even more oil deposits left. The writer offers three reasons why the world is not running of oil and probably will not in the near future. First, he states that oil being a known resource means that it is â€Å"not a finite entity† (125). He explains that the fact that man does not know all oil fields and hence needs to keep exploring for the same means that there is a probability that he will keep finding new fields with new oil. The second reason pointed by this author is that humankind has become better at exploring and exploiting as demand for resources rise. He explains that new technology has been a major contribution to the extraction of already existing oil fields. In addition, technological innovations are playing a major role in tracing new oil fields, and also enabling oil companies to exploit fields that had previously too difficult or expensive to exploit. The third reason cited by Lomborg is man’s ability to substitute. In his explanation, he argues that man is not specifically interested in oil for the sake of it, but his interest lies in the services that oil provides. This means that he can easily substitute to other sources of fuel, energy and heating when oil becomes too expensive. This then means that oil will only be used for services that cannot be easily be substituted. Sah, S. L. Encyclopedia of petroleum Science Engineering. New Delhi: Gyan Publishing House, 2003. Print In pages 86-87 of this book, Sah documents the significance of the discovery of giant oil fields in different parts of the world. Most importantly, he lists the importance that oil explorers attach to new oil wells. First on the list is the fact that the new fields contribute significantly to the world’s oil reserves. More to this, there is an understanding among geologists that the position of â€Å"tectonic setting, geological history and conditions for hydrocarbon formation†¦Ã¢â‚¬  (183) contribute significantly to the understanding of oi l origin and supply of the same in future. Second on Sah’s list is the fact that the giant fields are not located in one part of the world. It is however notable two thirds of such well are to be found in the Middle East. Third, Sah notes that even though the frequency of giant wells have decreased over the years, some regions like West Africa and Brazil which were previously unexplored continue to offer new prospects to the world oil market. Sah argues that the equation to present oil reserves is attained by adding past reserves to additions made to reserves and then subtracting the production made from such reserves (246). From this equation, he argues that oil reserves and their capacities are bound to change with time especially because new fields keep on being discovered; reservoirs fields keep on being extended; and changes in extraction technology keep presenting extractors with new ways of getting more oil from existing fields. Tertzakian, Peter. A thousand Barrels a Second: The coming oil break point and the challenges facing an energy dependent world. London: McGraw Hill professional, 2007. Print In the first chapter of his book titled â€Å"lighting the last whale lamp†, Tertzakian states categorically that the world is simply not running out of oil. He however observes that the world may be experiencing the cheap oil that is most preferred due to its low sulfur content. According to the writer, the dependency that people place on the cheap oil has grown tremendously over the years and is also facing pressures from forces from business, policy, geopolitical and environmental quotas, which may eventually lead to the growth and dependency reaching a breaking point. The writer is however quick to state that with radical technologies in oil exploration today, this will eventually lead to the rebalancing of oil production, which will in-turn lead to more growth in dependency on the same. Tertzakian notes that any time in the oil production that the world is faced with uncertainties, radical technologies are developed with contribute significantly to a rebalance in production. He however notes that at some point governments have had to impose aggressive taxes on oil products in order to rebalance demand, but this usually happens for a short-term before a solution to the oil production is found. Although this book chapter deals primarily with the whale oil which was sought after for lighting before the invention of kerosene and later the light bulb, Tertzakian draws similarities in the fact that the pressures that face human kind for energy solutions will always lead to greater inventions, which in turn mean that explorers will only give up exploring for oil when it has been completely proven that oil deposits are no more. This is unlikely to happen in the foreseeable future. Maugeri, Leornado. â€Å"Oil: Never Cry Wolf—Why The Petroleum Age Is Far From Over.† Science 204. 5674 (2004):1114-1115. This auth or argues that the Hubert model which oil doomsters are using to herald an end to oil is nothing more that geological faith. Maugeri observes that there are no enough geological facts for the oil doomsters to claim that they can specifically predict when the oil resources will cease from being. Although he specifically acknowledges the view that hydrocarbon reserves are finite, he says that it is contrary to the Hubert model for geologists or environmentalists to claim they know just how finite the oil reserves are. Like other authors analyzed herein, Maugeri draws a distinction between the already identified oil resources and the yet-to-be-discovered resources. He argues that besides the geological knowledge that most of the oil doomsters possess, they need also to acknowledge that technological development and economics of oil have taken over the evolution of oil production. Maugeri concludes by stating that â€Å"geology is not destiny, but rather only a part of a much complex p icture that does not indicate the world is running out of oil† (1115). This research paper on Is The World Running Out Of Oil? was written and submitted by user Rhett A. to help you with your own studies. You are free to use it for research and reference purposes in order to write your own paper; however, you must cite it accordingly. You can donate your paper here.

The Battle of Sexes in Society Essays

The Battle of Sexes in Society Essays The Battle of Sexes in Society Essay The Battle of Sexes in Society Essay The Battle of Sexes in Society Men have always had things easier in society compared to women. Women have had to work their way up to be the way it is in today’s society. However, unlike men, women always seem to get away with crimes or injustices. In the beginning of our nation, when the Declaration of Independence, as well as other important documents from the same time period, women were never really mentioned. The only humans that were important in society were the white men. The woman’s primary role was to cook, clean, and take care of the children. However, after many protests, women finally won over the right to vote, which is now stated in the 19th Amendment, in August of 1920. Slowly but surely, women also worked their way up the business ladder, especially in the medical field. They first began to really be noticed through midwifery, since women were more comfortable with other women during the birthing process. Men are always the primary figure in all areas of the human life; not just in society, but also in religion. They have always had the right to say anything, over women, and had never had to work to be able to have more opportunities available to them, unless it was through the social class ladder. When the Declaration of Independence was written the only figure of humankind that was on the creators mind, was the white male. All the rights for humanity in the United States were being created for white men. Slaves and women were just followers to the system of which whatever the white men decided, that’s how life would roll. A great example of how women can however get away with crimes or injustices comes from an earlier reading, Creepin’ While Your Sleepin’. It explains how a woman can play an innocent card just because of the sex they are. Most police officers would let any woman get away with an act of injustice, because it is not thought of a woman to do such a crime. With time, the battle of the sexes will fade, and both sexes will live with some kind of equality. However, this might just be the dream of an everyday dreamer. Throughout history, humankind is always at battle competing for something, or trying to be better than the other. This may be an issue or dream that will never be fulfilled, because the male will always want to be with more power and strength compared to a woman.

Definition and Examples of Short Answers in English

Definition and Examples of Short Answers in English In spoken English and informal writing, a short answer is a response made up of a subject and an auxiliary verb or modal. A short answer is generally regarded as more polite than just an abrupt yes or no. Conventionally, the verb in a short answer is in the same tense as the verb in the question. Also, the verb in the short answer should agree in person and number with its subject. Examples and Observations How did she do in her exams? Maria had already told me she had done quite well, but I was now flailing around to keep the conversation going.She passed.She is all right, isnt she?Yes, she is, he replied firmly.(Vikram Seth, An Equal Music. Random House, 1999)The poor lass took quite a fall, didnt she? Gelfrid remarked. Is she usually so clumsy?No, she isnt,† Judith answered.(Julie Garwood, The Secret. Pocket Books, 1992)Youre asking yourself, Can I give this child the best possible upbringing and keep her out of harms way her whole life long? The answer is no, you cant.(Barbara Kingsolver, The Bean Trees. Harper Row, 1988)Can we change? Yes, we can. Can they change? Yes, they can.(Oz Clarke, Oz Clarkes Pocket Wine Guide. Sterling, 2009)Will, youve been in love before, havent you? I mean, with Anna, of course . . . and your various . . . well, you have, havent you?Will looked into his glass. No. No, I havent.(Jennifer Donnelly, The Tea Rose. Macmillan, 2004) Whats up with him?His stomach is sick. Hes nervous about his speech.Hes got food poisoning! Helen declared. â€Å"Hasnt he?†Ã¢â‚¬Å"No, he has not!†Ã¢â‚¬Å"Yes, he has.†Ã¢â‚¬Å"No, he has not!†Ã¢â‚¬Å"Yes, he has.†(Marian Keyes, Anybody Out There? William Morrow, 2006)No, I wont, Jeremiahno I wontno I wont!I wont go, Ill stay here. Ill hear all I dont know, and say all I know. I will, at last, if I die for it. I will, I will, I will, I will!(Charles Dickens, Little Dorrit, 1857) Short-Answer Patterns Answers are often grammatically incomplete, because they do not need to repeat words that have just been said. A typical short answer pattern is subject auxiliary verb, together with whatever other words are really necessary. Can he swim? Yes, he can.(More natural than Yes, he can swim.)Has it stopped raining? No, it hasnt.Are you enjoying yourself? I certainly am.Youll be on holiday soon. Yes, I will.Dont forget to telephone. I wont.You didnt phone Debbie last night. No, but I did this morning. Non-auxiliary be and have are also used in short answers. Is she happy? I think she is.Have you a light? Yes, I have. We use do and did in answers to sentences that have neither an auxiliary verb nor non-auxiliary be or have. She likes cakes. She really does.That surprised you. It certainly did. Short answers can be followed by tags . . .. Nice day. Yes, it is, isnt it? Note that stressed, non-contracted forms are used in short answers.(Michael Swan, Practical English Usage. Oxford University Press, 1995) Short Answers With So, Neither, and Nor Sometimes a statement about one person also applies to another person. When this is the case, you can use a short answer with so for positive statements, and with neither or nor for negative statements using the same verb that was used in the statement. You use so, neither, or nor with an auiliary, modal, or the main verb be. The verb comes before the subject. You were different then. So were you.I dont normally drink at lunch. Neither do I.I cant do it. Nor can I. You can use not either instead of neither, in which case the verb comes after the subject. He doesnt understand. We dont either. You often use so in short answers after verbs such as think, hope, expect, imagine, and suppose, when you think that the answer to the question is yes. Youll be home at six? I hope so.So it was worth doing? I suppose so. You use Im afraid so when you are sorry that the answer is yes. Is it raining? Im afraid so. With suppose, think, imagine, or expect in short answers, you also form negatives with so. Will I see you again? I dont suppose so.Is Barry Knight a golfer? No, I dont think so. However, you say I hope not and Im afraid not. It isnt empty, is it? I hope not. (Collins COBUILD Active English Grammar. HarperCollins, 2003)

Bioprocessing requirements to manufacture a specific product Essay

Bioprocessing requirements to manufacture a specific product - Essay Example In lactic acid fermentation, a single molecule of pyruvate is changed into lactate. In the same process, ethanol and carbon dioxide are also resulting products. This kind of fermentation occurs in muscles of animals when the energy requirement exceeds the oxygen supply. This anaerobic process occurs, providing the organism with the energy required, in an anaerobic manner. Before this process can occur, though, a molecule of glucose has to be split into two molecules of pyruvate. This is a process referred to as glycolysis. In homolactic fermentation, two moles of lactic acid are anaerobically formed from a single molecule of glucose. When lactic acid is produced during fermentation, a racemic mixture of its two isomers is obtained. The L(+) and D(-) isomers will exist in equal quantities. When carbohydrate is fermented, glycolic and lactic acids will be found in the product mixture. This is the simplest form of fermentation. Lactic acid fermentation is a redox reaction that occurs in anaerobic conditions (Dworkin, 2006, 539). Lactic acid fermentation as a process finds use in the food industry since it is used in the production of yoghurt. Milk is fermented with bacteria that are harmless, mainly Streptococcus thermophilus and Lactobacillus bulgaricus. Milk is used as the culture in this process. When the pH of the milk is decreased, it congeals. The bacteria are responsible for producing compounds that give the resultant product the distinctive taste of yoghurt. By lowering the pH, the conditions become unfavourable for harmful bacteria, making this process effective. This process also finds use in producing sauerkraut. This process utilizes genus Leuconostoc (Dworkin, 2006, 541). In lactic acid fermentation, the product will be dependent on the strain of microbe used. The strain of microbe that is used for fermentation will influence the stereo-specificity of the product. The

Interracial Marriage in China Essay Example | Topics and Well Written Essays - 1500 words

Interracial Marriage in China - Essay Example The paper will then conclude by discussing the significance of race within this period. Branding, which is the word used within this text to describe the ultimate affects of any such a union, originated as the act of using an extremely hot implement to mark the flesh of humans and livestock. The practice of branding, including that of physically marking human beings, began long before recorded history. Although originally intended to mark ownership in cattle, it was also used as a form of punishment, and employed to distinguish criminals, slaves, and prisoners of war. In Western civilization, branding was banned as a form of punishment in the late 1800s.2 Historically, having a brand was a symbol or mark of identification and ownership - it labeled the person, significating what they were, rather than who they were; criminal, slave, and so forth, and the fact of being branded considered a sign of public shame and personal humiliation. Of course, this marking, including its accompanying reputation, remained with a person for the rest of their life. The use of this word in the above statement, therefore, gives a very clear indication of the attitudes among the general population of the early 20th century towards the idea of interracial marriages. Not only were these types of relationships negatively viewed, but they were also considered as being both shameful and humiliating. Consequently, Tom Frew, his Chinese wife, and their unborn children, would have all been branded with this public mark of shame, of derision, and of exclusion in certain quarters. Early Twentieth Century Worldviews "It has been justly remarked that a nation's civilization may be estimated by the rank which females hold in society. If the civilization of China be judged of by this test, she is surely far from occupying that first place which she so strongly claims" (Chinese Repository, vol. 2, 1833, p. 313). Although this quote is earlier than the period that this essay is addressing, existing worldviews during the earlier twentieth century were based on similar principles, and were rooted within this theory. Within the 1920s, Chinese women were generally seen throughout the Western world, as demonstrated through the above quote, as representative objects of an inferior, underdeveloped society. Almost despised, considered second-rate, this very common worldview, meant that Tom Frew and his Chinese family would face severe discrimatory attitudes and practices where ever they chose to live. "Could he live anywhere, with a Chinese wife" the statement demands. Only, it would seem, with the greatest of difficulties. America, in the early twentieth century, was very suspicious of the Chinese, especially in relation to the influx of immigration. Chinatowns had sprung up throughout many of the major cities - the result of the clash of two distinctively different cultures - and although living in the same city, there was little attempt at integration by either the Chinese or American people. Each community kept within its cultural boundaries, holding onto its group identity, and ensuring the continuation of this through imposing restrictive

Warren G. Harding, President Essay -- Politics

Warren G. Harding, President (1865-1923) Harding was born on November 2, 1865, in Corsica (now Bloomington Grove), Ohio. He was eldest of eight children. His father, George Tryon Harding, was a farmer and a doctor. His mother, Phoebe Dickerson Harding, was a "gentle, pious" woman who devoted herself to her children. As a boy Warren helped his fater on the farm. In the summer he worked in a sawmill that made brooms, and he drove a team of horses for the Toledo and Ohio Central Railroaad. His father was later quoted as saying , "Warren was always willing to work hard if there was any money in it." Later Warren would become a printers apprentance, and office boy on the Caledonia Argus, a local newspaper. There he learned how to set type and gained his first newspaper experience. In 1879, at the age of 14, Harding entered Ohio Central College in Iberia. After graduating in 1882 he took a job as a schoolteacher. But he gave it up after one term, calling it the hardest job he ever had. The following year the Hardings moved to Marion, Ohio. Harding studied law for a few months, bu...

Flow Induced Vibration

FLOW INDUCED VIBRATIONS IN PIPES, A FINITE ELEMENT APPROACH IVAN GRANT Bachelor of Science in Mechanical Engineering Nagpur University Nagpur, India June, 2006 submitted in partial ful? llment of requirements for the degree MASTERS OF SCIENCE IN MECHANICAL ENGINEERING at the CLEVELAND STATE UNIVERSITY May, 2010 This thesis has been approved for the department of MECHANICAL ENGINEERING and the College of Graduate Studies by: Thesis Chairperson, Majid Rashidi, Ph. D. Department & Date Asuquo B. Ebiana, Ph. D. Department & Date Rama S. Gorla, Ph. D. Department & Date ACKNOWLEDGMENTS I would like to thank my advisor Dr. Majid Rashidi and Dr.Paul Bellini, who provided essential support and assistance throughout my graduate career, and also for their guidance which immensely contributed towards the completion of this thesis. This thesis would not have been realized without their support. I would also like to thank Dr. Asuquo. B. Ebiana and Dr. Rama. S. Gorla for being in my thesis committe e. Thanks are also due to my parents,my brother and friends who have encouraged, supported and inspired me. FLOW INDUCED VIBRATIONS IN PIPES, A FINITE ELEMENT APPROACH IVAN GRANT ABSTRACT Flow induced vibrations of pipes with internal ? uid ? ow is studied in this work.Finite Element Analysis methodology is used to determine the critical ? uid velocity that induces the threshold of pipe instability. The partial di? erential equation of motion governing the lateral vibrations of the pipe is employed to develop the sti? ness and inertia matrices corresponding to two of the terms of the equations of motion. The Equation of motion further includes a mixed-derivative term that was treated as a source for a dissipative function. The corresponding matrix with this dissipative function was developed and recognized as the potentially destabilizing factor for the lateral vibrations of the ? id carrying pipe. Two types of boundary conditions, namely simply-supported and cantilevered were consi dered for the pipe. The appropriate mass, sti? ness, and dissipative matrices were developed at an elemental level for the ? uid carrying pipe. These matrices were then assembled to form the overall mass, sti? ness, and dissipative matrices of the entire system. Employing the ? nite element model developed in this work two series of parametric studies were conducted. First, a pipe with a constant wall thickness of 1 mm was analyzed. Then, the parametric studies were extended to a pipe with variable wall thickness.In this case, the wall thickness of the pipe was modeled to taper down from 2. 54 mm to 0. 01 mm. This study shows that the critical velocity of a pipe carrying ? uid can be increased by a factor of six as the result of tapering the wall thickness. iv TABLE OF CONTENTS ABSTRACT LIST OF FIGURES LIST OF TABLES I INTRODUCTION 1. 1 1. 2 1. 3 1. 4 II Overview of Internal Flow Induced Vibrations in Pipes . . . . . . Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Composition of Thesis . . . . . . . . . . . . . . . . . . . . . . . iv vii ix 1 1 2 2 3 FLOW INDUCED VIBRATIONS IN PIPES, A FINITE ELEMENT APPROACH 2. 1 Mathematical Modelling . . . . . . . . . . . . . . . . . . . . . . . 2. 1. 1 2. 2 Equations of Motion . . . . . . . . . . . . . . . . . . . 4 4 4 12 12 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . 2. 2. 1 2. 2. 2 2. 2. 3 Shape Functions . . . . . . . . . . . . . . . . . . . . . Formulating the Sti? ness Matrix for a Pipe Carrying Fluid 14 Forming the Matrix for the Force that conforms the Fluid to the Pipe . . . . . . . . . . . . . . . . . . . . . 21 2. 2. 4 2. 2. 5Dissipation Matrix Formulation for a Pipe carrying Fluid 26 Inertia Matrix Formulation for a Pipe carrying Fluid . 28 III FLOW INDUCED VIBRATIONS IN PIPES, A FINITE ELEMENT APPROACH 31 v 3. 1 Forming Global Sti? ness Matrix from Elemental Sti? ness Matrices . . . . . . . . . . . . . . . . . . . . 31 3. 2 Applying Boundary Conditions to Global Sti? ness Matrix for simply supported pipe with ? uid ? ow . . . . 33 3. 3 Applying Boundary Conditions to Global Sti? ness Matrix for a cantilever pipe with ? uid ? ow . . . . . . . 34 3. 4 MATLAB Programs for Assembling Global Matrices for Simply Supported and Cantilever pipe carrying ? uid . . . . . . . . . . 35 35 36 3. 5 3. 6 MATLAB program for a simply supported pipe carrying ? uid . . MATLAB program for a cantilever pipe carrying ? uid . . . . . . IV FLOW INDUCED VIBRATIONS IN PIPES, A FINITE ELEMENT APPROACH 4. 1 V Parametric Study . . . . . . . . . . . . . . . . . . . . . . . . . . 37 37 FLOW INDUCED VIBRATIONS IN PIPES, A FINITE ELEMENT APPROACH 5. 1 Tapered Pipe Carrying Fluid . . . . . . . . . . . . . . . . . . . . 42 42 47 50 50 51 54 MATLAB program for Simply Supported Pipe Carrying Fluid . . MATLAB Program for Cantilever Pipe Carrying Fluid . . . . . . MATLAB Program for Tapered Pipe Carrying Flu id . . . . . . 54 61 68 VI RESULTS AND DISCUSSIONS 6. 1 6. 2 Contribution of the Thesis . . . . . . . . . . . . . . . . . . . . . Future Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BIBLIOGRAPHY Appendices 0. 1 0. 2 0. 3 vi LIST OF FIGURES 2. 1 2. 2 Pinned-Pinned Pipe Carrying Fluid * . . . . . . . . . . . . . . Pipe Carrying Fluid, Forces and Moments acting on Elements (a) Fluid (b) Pipe ** . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 7 9 10 11 13 14 15 16 17 21 33 34 36 2. 3 2. 4 2. 5 2. 6 2. 7 2. 8 2. 9 Force due to Bending . . . . . . . . . . . . . . . . . . . . . . . . .Force that Conforms Fluid to the Curvature of Pipe . . . . . Coriolis Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inertia Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pipe Carrying Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . Beam Element Model . . . . . . . . . . . . . . . . . . . . . . . . . Relationship between Stress and Stra in, Hooks Law . . . . . . 2. 10 Plain sections remain plane . . . . . . . . . . . . . . . . . . . . . 2. 11 Moment of Inertia for an Element in the Beam . . . . . . . . . 2. 12 Pipe Carrying Fluid Model . . . . . . . . . . . . . . . . . . . . . 3. 1 3. 2 3. 4. 1 Representation of Simply Supported Pipe Carrying Fluid . . Representation of Cantilever Pipe Carrying Fluid . . . . . . . Pinned-Free Pipe Carrying Fluid* . . . . . . . . . . . . . . . . . Reduction of Fundamental Frequency for a Pinned-Pinned Pipe with increasing Flow Velocity . . . . . . . . . . . . . . . . 4. 2 Shape Function Plot for a Cantilever Pipe with increasing Flow Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. 3 Reduction of Fundamental Frequency for a Cantilever Pipe with increasing Flow Velocity . . . . . . . . . . . . . . . . . . . . 5. 1 Representation of Tapered Pipe Carrying Fluid . . . . . . . 39 40 41 42 vii 5. 2 6. 1 Introducing a Taper in the Pipe Carrying Fluid . . . . . . . . Representation of Pipe Carrying Fluid and Tapered Pipe Carrying Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 47 viii LIST OF TABLES 4. 1 Reduction of Fundamental Frequency for a Pinned-Pinned Pipe with increasing Flow Velocity . . . . . . . . . . . . . . . . 38 4. 2 Reduction of Fundamental Frequency for a Pinned-Free Pipe with increasing Flow Velocity . . . . . . . . . . . . . . . . . . . . 40 5. 1 Reduction of Fundamental Frequency for a Tapered pipe with increasing Flow Velocity . . . . . . . . . . . . . . . . . . . . . . 46 6. 1 Reduction of Fundamental Frequency for a Tapered Pipe with increasing Flow Velocity . . . . . . . . . . . . . . . . . . . . . . . 48 6. 2 Reduction of Fundamental Frequency for a Pinned-Pinned Pipe with increasing Flow Velocity . . . . . . . . . . . . . . . . 49 ix CHAPTER I INTRODUCTION 1. 1 Overview of Internal Flow Induced Vibrations in Pipes The ? ow of a ? uid through a pipe can impose pressures on the walls of the pipe c ausing it to de? ect under certain ? ow conditions. This de? ection of the pipe may lead to structural instability of the pipe.The fundamental natural frequency of a pipe generally decreases with increasing velocity of ? uid ? ow. There are certain cases where decrease in this natural frequency can be very important, such as very high velocity ? ows through ? exible thin-walled pipes such as those used in feed lines to rocket motors and water turbines. The pipe becomes susceptible to resonance or fatigue failure if its natural frequency falls below certain limits. With large ? uid velocities the pipe may become unstable. The most familiar form of this instability is the whipping of an unrestricted garden hose.The study of dynamic response of a ? uid conveying pipe in conjunction with the transient vibration of ruptured pipes reveals that if a pipe ruptures through its cross section, then a ? exible length of unsupported pipe is left spewing out ? uid and is free to whip about and im pact other structures. In power plant plumbing pipe whip is a possible mode of failure. A 1 2 study of the in? uence of the resulting high velocity ? uid on the static and dynamic characteristics of the pipes is therefore necessary. 1. 2 Literature Review Initial investigations on the bending vibrations of a simply supported pipe containing ? id were carried out by Ashley and Haviland[2]. Subsequently,Housner[3] derived the equations of motion of a ? uid conveying pipe more completely and developed an equation relating the fundamental bending frequency of a simply supported pipe to the velocity of the internal ? ow of the ? uid. He also stated that at certain critical velocity, a statically unstable condition could exist. Long[4] presented an alternate solution to Housner’s[3] equation of motion for the simply supported end conditions and also treated the ? xed-free end conditions. He compared the analysis with experimental results to con? rm the mathematical model.His experi mental results were rather inconclusive since the maximum ? uid velocity available for the test was low and change in bending frequency was very small. Other e? orts to treat this subject were made by Benjamin, Niordson[6] and Ta Li. Other solutions to the equations of motion show that type of instability depends on the end conditions of the pipe carrying ? uid. If the ? ow velocity exceeds the critical velocity pipes supported at both ends bow out and buckle[1]. Straight Cantilever pipes fall into ? ow induced vibrations and vibrate at a large amplitude when ? ow velocity exceeds critical velocity[8-11]. . 3 Objective The objective of this thesis is to implement numerical solutions method, more specifically the Finite Element Analysis (FEA) to obtain solutions for di? erent pipe con? gurations and ? uid ? ow characteristics. The governing dynamic equation describing the induced structural vibrations due to internal ? uid ? ow has been formed and dis- 3 cussed. The governing equatio n of motion is a partial di? erential equation that is fourth order in spatial variable and second order in time. Parametric studies have been performed to examine the in? uence of mass distribution along the length of the pipe carrying ? id. 1. 4 Composition of Thesis This thesis is organized according to the following sequences. The equations of motions are derived in chapter(II)for pinned-pinned and ? xed-pinned pipe carrying ? uid. A ? nite element model is created to solve the equation of motion. Elemental matrices are formed for pinned-pinned and ? xed-pinned pipe carrying ? uid. Chapter(III)consists of MATLAB programs that are used to assemble global matrices for the above cases. Boundary conditions are applied and based on the user de? ned parameters fundamental natural frequency for free vibration is calculated for various pipe con? urations. Parametric studies are carried out in the following chapter and results are obtained and discussed. CHAPTER II FLOW INDUCED VIBRATION S IN PIPES, A FINITE ELEMENT APPROACH In this chapter,a mathematical model is formed by developing equations of a straight ? uid conveying pipe and these equations are later solved for the natural frequency and onset of instability of a cantilever and pinned-pinned pipe. 2. 1 2. 1. 1 Mathematical Modelling Equations of Motion Consider a pipe of length L, modulus of elasticity E, and its transverse area moment I. A ? uid ? ows through the pipe at pressure p and density ? t a constant velocity v through the internal pipe cross-section of area A. As the ? uid ? ows through the de? ecting pipe it is accelerated, because of the changing curvature of the pipe and the lateral vibration of the pipeline. The vertical component of ? uid pressure applied to the ? uid element and the pressure force F per unit length applied on the ? uid element by the tube walls oppose these accelerations. Referring to ? gures (2. 1) and 4 5 Figure 2. 1: Pinned-Pinned Pipe Carrying Fluid * (2. 2),balancing the forces in the Y direction on the ? uid element for small deformations, gives F ? A ? ? ? 2Y = ? A( + v )2 Y ? x2 ? t ? x (2. 1) The pressure gradient in the ? uid along the length of the pipe is opposed by the shear stress of the ? uid friction against the tube walls. The sum of the forces parallel Figure 2. 2: Pipe Carrying Fluid, Forces and Moments acting on Elements (a) Fluid (b) Pipe ** to the pipe axis for a constant ? ow velocity gives 0 0 * Flow Induced Vibrations,Robert D. Blevins,Krieger. 1977,P 289 ** Flow Induced Vibrations,Robert D. Blevins,Krieger. 1977,P 289 6 A ?p + ? S = 0 ? x (2. 2) Where S is the inner perimeter of the pipe, and ? s the shear stress on the internal surface of the pipe. The equations of motions of the pipe element are derived as follows. ?T ? 2Y + ? S ? Q 2 = 0 ? x ? x (2. 3) Where Q is the transverse shear force in the pipe and T is the longitudinal tension in the pipe. The forces on the element of the pipe normal to the pipe axis accelerate the pi pe element in the Y direction. For small deformations, ? 2Y ? 2Y ? Q +T 2 ? F =m 2 ? x ? x ? t (2. 4) Where m is the mass per unit length of the empty pipe. The bending moment M in the pipe, the transverse shear force Q and the pipe deformation are related by ? 3Y ?M = EI 3 ? x ? x Q=? (2. 5) Combining all the above equations and eliminating Q and F yields: EI ? 4Y ? 2Y ? ? ? Y + (? A ? T ) 2 + ? A( + v )2 Y + m 2 = 0 4 ? x ? x ? t ? x ? t (2. 6) The shear stress may be eliminated from equation 2. 2 and 2. 3 to give ? (? A ? T ) =0 ? x (2. 7) At the pipe end where x=L, the tension in the pipe is zero and the ? uid pressure is equal to ambient pressure. Thus p=T=0 at x=L, ? A ? T = 0 (2. 8) 7 The equation of motion for a free vibration of a ? uid conveying pipe is found out by substituting ? A ? T = 0 from equation 2. 8 in equation 2. 6 and is given by the equation 2. EI ? 2Y ? 2Y ? 4Y ? 2Y +M 2 =0 + ? Av 2 2 + 2? Av ? x4 ? x ? x? t ? t (2. 9) where the mass per unit length of the pi pe and the ? uid in the pipe is given by M = m + ? A. The next section describes the forces acting on the pipe carrying ? uid for each of the components of eq(2. 9) Y F1 X Z EI ? 4Y ? x4 Figure 2. 3: Force due to Bending Representation of the First Term in the Equation of Motion for a Pipe Carrying Fluid 8 The term EI ? Y is a force component acting on the pipe as a result of bending of ? x4 the pipe. Fig(2. 3) shows a schematic view of this force F1. 4 9 Y F2 X Z ?Av 2 ? 2Y ? x2 Figure 2. : Force that Conforms Fluid to the Curvature of Pipe Representation of the Second Term in the Equation of Motion for a Pipe Carrying Fluid The term ? Av 2 ? Y is a force component acting on the pipe as a result of ? ow ? x2 around a curved pipe. In other words the momentum of the ? uid is changed leading to a force component F2 shown schematically in Fig(2. 4) as a result of the curvature in the pipe. 2 10 Y F3 X Z 2? Av ? 2Y ? x? t Figure 2. 5: Coriolis Force Representation of the Third Term in t he Equation of Motion for a Pipe Carrying Fluid ? Y The term 2? Av ? x? t is the force required to rotate the ? id element as each point 2 in the span rotates with angular velocity. This force is a result of Coriolis E? ect. Fig(2. 5) shows a schematic view of this force F3. 11 Y F4 X Z M ? 2Y ? t2 Figure 2. 6: Inertia Force Representation of the Fourth Term in the Equation of Motion for a Pipe Carrying Fluid The term M ? Y is a force component acting on the pipe as a result of Inertia ? t2 of the pipe and the ? uid ? owing through it. Fig(2. 6) shows a schematic view of this force F4. 2 12 2. 2 Finite Element Model Consider a pipeline span that has a transverse de? ection Y(x,t) from its equillibrium position.The length of the pipe is L,modulus of elasticity of the pipe is E,and the area moment of inertia is I. The density of the ? uid ? owing through the pipe is ? at pressure p and constant velocity v,through the internal pipe cross section having area A. Flow of the ? uid through the de? ecting pipe is accelerated due to the changing curvature of the pipe and the lateral vibration of the pipeline. From the previous section we have the equation of motion for free vibration of a ? uid convering pipe: EI ? 2Y ? 2Y ? 2Y ? 4Y + ? Av 2 2 + 2? Av +M 2 =0 ? x4 ? x ? x? t ? t (2. 10) 2. 2. 1 Shape Functions The essence of the ? ite element method,is to approximate the unknown by an expression given as n w= i=1 Ni ai where Ni are the interpolating shape functions prescribed in terms of linear independent functions and ai are a set of unknown parameters. We shall now derive the shape functions for a pipe element. 13 Y R R x L2 L L1 X Figure 2. 7: Pipe Carrying Fluid Consider an pipe of length L and let at point R be at distance x from the left end. L2=x/L and L1=1-x/L. Forming Shape Functions N 1 = L12 (3 ? 2L1) N 2 = L12 L2L N 3 = L22 (3 ? 2L2) N 4 = ? L1L22 L Substituting the values of L1 and L2 we get (2. 11) (2. 12) (2. 13) (2. 14) N 1 = (1 ? /l)2 (1 + 2x/l) N 2 = (1 ? x/l)2 x/l N 3 = (x/l)2 (3 ? 2x/l) N 4 = ? (1 ? x/l)(x/l)2 (2. 15) (2. 16) (2. 17) (2. 18) 14 2. 2. 2 Formulating the Sti? ness Matrix for a Pipe Carrying Fluid ?1 ?2 W1 W2 Figure 2. 8: Beam Element Model For a two dimensional beam element, the displacement matrix in terms of shape functions can be expressed as ? ? w1 ? ? ? ? ? ?1 ? ? ? [W (x)] = N 1 N 2 N 3 N 4 ? ? ? ? ? w2? ? ? ?2 (2. 19) where N1, N2, N3 and N4 are the displacement shape functions for the two dimensional beam element as stated in equations (2. 15) to (2. 18). The displacements and rotations at end 1 is given by w1, ? and at end 2 is given by w2 , ? 2. Consider the point R inside the beam element of length L as shown in ? gure(2. 7) Let the internal strain energy at point R is given by UR . The internal strain energy at point R can be expressed as: 1 UR = ? 2 where ? is the stress and is the strain at the point R. (2. 20) 15 ? E 1 ? Figure 2. 9: Relationship between Stress and Strain, Hooks Law Also; ? =E Rel ation between stress and strain for elastic material, Hooks Law Substituting the value of ? from equation(2. 21) into equation(2. 20) yields 1 UR = E 2 (2. 21) 2 (2. 22) 16 A1 z B1 w A z B u x Figure 2. 0: Plain sections remain plane Assuming plane sections remain same, = du dx (2. 23) (2. 24) (2. 25) u=z dw dx d2 w =z 2 dx To obtain the internal energy for the whole beam we integrate the internal strain energy at point R over the volume. The internal strain energy for the entire beam is given as: UR dv = U vol (2. 26) Substituting the value of from equation(2. 25) into (2. 26) yields U= vol 1 2 E dv 2 (2. 27) Volume can be expressed as a product of area and length. dv = dA. dx (2. 28) 17 based on the above equation we now integrate equation (2. 27) over the area and over the length. L U= 0 A 1 2 E dAdx 2 (2. 29) Substituting the value of rom equation(2. 25) into equation (2. 28) yields L U= 0 A 1 d2 w E(z 2 )2 dAdx 2 dx (2. 30) Moment of Inertia I for the beam element is given as = dA z Figure 2. 11: Moment of Inertia for an Element in the Beam I= z 2 dA (2. 31) Substituting the value of I from equation(2. 31) into equation(2. 30) yields L U = EI 0 1 d2 w 2 ( ) dx 2 dx2 (2. 32) The above equation for total internal strain energy can be rewritten as L U = EI 0 1 d2 w d2 w ( )( )dx 2 dx2 dx2 (2. 33) 18 The potential energy of the beam is nothing but the total internal strain energy. Therefore, L ? = EI 0 1 d2 w d2 w ( )( )dx 2 dx2 dx2 (2. 34)If A and B are two matrices then applying matrix property of the transpose, yields (AB)T = B T AT (2. 35) We can express the Potential Energy expressed in equation(2. 34) in terms of displacement matrix W(x)equation(2. 19) as, 1 ? = EI 2 From equation (2. 19) we have ? ? w1 ? ? ? ? ? ?1 ? ? ? [W ] = N 1 N 2 N 3 N 4 ? ? ? ? ? w2? ? ? ?2 ? ? N1 ? ? ? ? ? N 2? ? ? [W ]T = ? ? w1 ? 1 w2 ? 2 ? ? ? N 3? ? ? N4 L (W )T (W )dx 0 (2. 36) (2. 37) (2. 38) Substituting the values of W and W T from equation(2. 37) and equation(2. 3 8) in equation(2. 36) yields ? N1 ? ? ? N 2 ? w1 ? 1 w2 ? 2 ? ? ? N 3 ? N4 ? ? ? ? ? ? N1 ? ? ? ? ? w1 ? ? ? ? ?1 ? ? ? ? ? dx (2. 39) ? ? ? w2? ? ? ?2 1 ? = EI 2 L 0 N2 N3 N4 19 where N1, N2, N3 and N4 are the displacement shape functions for the two dimensional beam element as stated in equations (2. 15) to (2. 18). The displacements and rotations at end 1 is given by w1, ? 1 and at end 2 is given by w2 , ? 2. 1 ? = EI 2 L 0 (N 1 ) ? ? ? N 2 N 1 ? w1 ? 1 w2 ? 2 ? ? ? N 3 N 1 ? N4 N1 ? 2 N1 N2 (N 2 )2 N3 N2 N4 N2 N1 N3 N2 N3 (N 3 )2 N4 N3 N1 N4 N2 N4 N3 N4 (N 4 )2 ? w1 ? ? ? ? ? 1 ? ? ? ? ? dx ? ? ?w2? ? ? 2 (2. 40) where ? 2 (N 1 ) ? ? L ? N 2 N 1 ? [K] = ? 0 ? N 3 N 1 ? ? N4 N1 N1 N2 (N 2 )2 N3 N2 N4 N2N1 N3 N2 N3 (N 3 ) 2 N1 N4 ? N4 N3 ? ? N2 N4 ? ? ? dx ? N3 N4 ? ? 2 (N 4 ) (2. 41) N 1 = (1 ? x/l)2 (1 + 2x/l) N 2 = (1 ? x/l)2 x/l N 3 = (x/l)2 (3 ? 2x/l) N 4 = ? (1 ? x/l)(x/l)2 (2. 42) (2. 43) (2. 44) (2. 45) The element sti? ness matrix for the beam is obtained by substit uting the values of shape functions from equations (2. 42) to (2. 45) into equation(2. 41) and integrating every element in the matrix in equation(2. 40) over the length L. 20 The Element sti? ness matrix for a beam element; ? ? 12 6l ? 12 6l ? ? ? ? 2 2? 4l ? 6l 2l ? EI ? 6l ? [K e ] = 3 ? ? l 12 ? 6l 12 ? 6l? ? ? ? ? 2 2 6l 2l ? 6l 4l (2. 46) 1 2. 2. 3 Forming the Matrix for the Force that conforms the Fluid to the Pipe A X ? r ? _______________________ x R Y Figure 2. 12: Pipe Carrying Fluid Model B Consider a pipe carrying ? uid and let R be a point at a distance x from a reference plane AB as shown in ? gure(2. 12). Due to the ? ow of the ? uid through the pipe a force is introduced into the pipe causing the pipe to curve. This force conforms the ? uid to the pipe at all times. Let W be the transverse de? ection of the pipe and ? be angle made by the pipe due to the ? uid ? ow with the neutral axis. ? and ? represent the unit vectors along the X i j ? nd Y axis and r and ? rep resent the two unit vectors at point R along the r and ? ? ? axis. At point R,the vectors r and ? can be expressed as ? r = cos + sin ? i j (2. 47) ? ? = ? sin + cos i j Expression for slope at point R is given by; tan? = dW dx (2. 48) (2. 49) 22 Since the pipe undergoes a small de? ection, hence ? is very small. Therefore; tan? = ? ie ? = dW dx (2. 51) (2. 50) The displacement of a point R at a distance x from the reference plane can be expressed as; ? R = W ? + r? j r We di? erentiate the above equation to get velocity of the ? uid at point R ? ? ? j ? r ? R = W ? + r? + rr ? r = vf ? here vf is the velocity of the ? uid ? ow. Also at time t; r ? d? r= ? dt ie r ? d? d? = r= ? d? dt ? Substituting the value of r in equation(2. 53) yields ? ? ? ? j ? r R = W ? + r? + r (2. 57) (2. 56) (2. 55) (2. 53) (2. 54) (2. 52) ? Substituting the value of r and ? from equations(2. 47) and (2. 48) into equation(2. 56) ? yields; ? ? ? ?j ? R = W ? + r[cos + sin + r? [? sin + cos i j] i j] Sin ce ? is small The velocity at point R is expressed as; ? ? ? i ? j R = Rx? + Ry ? (2. 59) (2. 58) 23 ? ? i ? j ? ? R = (r ? r )? + (W + r? + r? )? ? ? The Y component of velocity R cause the pipe carrying ? id to curve. Therefore, (2. 60) 1 ? ? ? ? T = ? f ARy Ry (2. 61) 2 ? ? where T is the kinetic energy at the point R and Ry is the Y component of velocity,? f is the density of the ? uid,A is the area of cross-section of the pipe. ? ? Substituting the value of Ry from equation(2. 60) yields; 1 ? ? ? ? ? ? ? ? ? T = ? f A[W 2 + r2 ? 2 + r2 ? 2 + 2W r? + 2W ? r + 2rr ] 2 (2. 62) Substituting the value of r from equation(2. 54) and selecting the ? rst,second and the ? fourth terms yields; 1 2 ? ? T = ? f A[W 2 + vf ? 2 + 2W vf ? ] 2 (2. 63) Now substituting the value of ? from equation(2. 51) into equation(2. 3) yields; dW 2 dW dW 1 2 dW 2 ) + vf ( ) + 2vf ( )( )] T = ? f A[( 2 dt dx dt dx From the above equation we have these two terms; 1 2 dW 2 ? f Avf ( ) 2 dx 2? f Avf ( dW dW )( ) dt dx (2. 65) (2. 66) (2. 64) The force acting on the pipe due to the ? uid ? ow can be calculated by integrating the expressions in equations (2. 65) and (2. 66) over the length L. 1 2 dW 2 ? f Avf ( ) 2 dx (2. 67) L The expression in equation(2. 67) represents the force that causes the ? uid to conform to the curvature of the pipe. 2? f Avf ( L dW dW )( ) dt dx (2. 68) 24 The expression in equation(2. 68) represents the coriolis force which causes the ? id in the pipe to whip. The equation(2. 67) can be expressed in terms of displacement shape functions derived for the pipe ? =T ? V ? = L 1 2 dW 2 ? f Avf ( ) 2 dx (2. 69) Rearranging the equation; 2 ? = ? f Avf L 1 dW dW ( )( ) 2 dx dx (2. 70) For a pipe element, the displacement matrix in terms of shape functions can be expressed as ? ? w1 ? ? ? ? ? ?1 ? ? ? [W (x)] = N 1 N 2 N 3 N 4 ? ? ? ? ? w2? ? ? ?2 (2. 71) where N1, N2, N3 and N4 are the displacement shape functions pipe element as stated in equations (2. 15) to (2. 18). The displacements and rotations at end 1 is given by w1, ? 1 and at end 2 is given by w2 , ? . Refer to ? gure(2. 8). Substituting the shape functions determined in equations (2. 15) to (2. 18) ? ? N1 ? ? ? ? ? N 2 ? ? ? ? N1 w1 ? 1 w2 ? 2 ? ? ? N3 ? ? ? ? N4 ? ? w1 ? ? ? ? ? ?1 ? ? ? N 4 ? ? dx (2. 72) ? ? ? w2? ? ? ?2 L 2 ? = ? f Avf 0 N2 N3 25 L 2 ? = ? f Avf 0 (N 1 ) ? ? ? N 2 N 1 ? w1 ? 1 w2 ? 2 ? ? ? N 3 N 1 ? N4 N1 ? 2 N1 N2 (N 2 )2 N3 N2 N4 N2 N1 N3 N2 N3 (N 3 )2 N4 N3 N1 N4 N2 N4 N3 N4 (N 4 )2 ? w1 ? ? ? ? ? 1 ? ? ? ? ? dx ? ? ?w2? ? ? 2 (2. 73) where (N 1 ) ? ? L ? N 2 N 1 ? ? 0 ? N 3 N 1 ? ? N4 N1 ? 2 N1 N2 (N 2 )2 N3 N2 N4 N2 N1 N3 N2 N3 (N 3 ) 2 N1 N4 ? 2 [K2 ] = ? f Avf N4 N3 ? N2 N4 ? ? ? dx ? N3 N4 ? ? 2 (N 4 ) (2. 74) The matrix K2 represents the force that conforms the ? uid to the pipe. Substituting the values of shape functions equations(2. 15) to (2. 18) and integrating it over the length gives us the elemental matrix for the ? 36 3 ? 36 ? ? 4 ? 3 ? Av 2 ? 3 ? [K2 ]e = ? 30l 36 ? 3 36 ? ? 3 ? 1 ? 3 above force. ? 3 ? ? ? 1? ? ? ? ? 3? ? 4 (2. 75) 26 2. 2. 4 Dissipation Matrix Formulation for a Pipe carrying Fluid The dissipation matrix represents the force that causes the ? uid in the pipe to whip creating instability in the system. To formulate this matrix we recall equation (2. 4) and (2. 68) The dissipation function is given by; D= L 2? f Avf ( dW dW )( ) dt dx (2. 76) Where L is the length of the pipe element, ? f is the density of the ? uid, A area of cross-section of the pipe, and vf velocity of the ? uid ? ow. Recalling the displacement shape functions mentioned in equations(2. 15) to (2. 18); N 1 = (1 ? x/l)2 (1 + 2x/l) N 2 = (1 ? x/l)2 x/l N 3 = (x/l)2 (3 ? 2x/l) N 4 = ? (1 ? x/l)(x/l)2 (2. 77) (2. 78) (2. 79) (2. 80) The Dissipation Matrix can be expressed in terms of its displacement shape functions as shown in equations(2. 77) to (2. 80). ? ? N1 ? ? ? ? ? N 2 ? L ? ? D = 2? Avf ? N1 N2 N3 N4 w1 ? 1 w2 ? 2 ? ? ? 0 N3 ? ? ? ? N4 (N 1 ) ? ? ? N 2 N 1 ? w1 ? 1 w2 ? 2 ? ? ? N 3 N 1 ? N4 N1 ? 2 ? ? w1 ? ? ? ? ? ?1 ? ? ? ? ? dx ? ? ? w2? ? ? ?2 (2. 81) N1 N2 (N 2 )2 N3 N2 N4 N2 N1 N3 N2 N3 (N 3 )2 N4 N3 N1 N4 N2 N4 N3 N4 (N 4 )2 L 2? f Avf 0 ? w1 ? ? ? ? ? 1 ? ? ? ? ? dx ? ? ?w2? ? ? 2 (2. 82) 27 Substituting the values of shape functions from equations(2. 77) to (2. 80) and integrating over the length L yields; ? ? ? 30 6 30 ? 6 ? ? ? ? 0 6 ? 1? ?Av ? 6 ? ? [D]e = ? ? 30 30 ? 6 30 6 ? ? ? ? ? 6 1 ? 6 0 [D]e represents the elemental dissipation matrix. (2. 83) 28 2. 2. 5Inertia Matrix Formulation for a Pipe carrying Fluid Consider an element in the pipe having an area dA, length x, volume dv and mass dm. The density of the pipe is ? and let W represent the transverse displacement of the pipe. The displacement model for the Assuming the displacement model of the element to be W (x, t) = [N ]we (t) (2. 84) where W is the vector of displacements,[N] is the matrix of shape functions and we is the vecto r of nodal displacements which is assumed to be a function of time. Let the nodal displacement be expressed as; W = weiwt Nodal Velocity can be found by di? erentiating the equation() with time. W = (iw)weiwt (2. 86) (2. 85) Kinetic Energy of a particle can be expressed as a product of mass and the square of velocity 1 T = mv 2 2 (2. 87) Kinetic energy of the element can be found out by integrating equation(2. 87) over the volume. Also,mass can be expressed as the product of density and volume ie dm = ? dv T = v 1 ? 2 ? W dv 2 (2. 88) The volume of the element can be expressed as the product of area and the length. dv = dA. dx (2. 89) Substituting the value of volume dv from equation(2. 89) into equation(2. 88) and integrating over the area and the length yields; T = ? w2 2 ? ?W 2 dA. dx A L (2. 90) 29 ?dA = ?A A (2. 91) Substituting the value of A ?dA in equation(2. 90) yields; Aw2 2 T = ? W 2 dx L (2. 92) Equation(2. 92) can be written as; Aw2 2 T = ? ? W W dx L (2. 93) The Lagr ange equations are given by d dt where L=T ? V (2. 95) ? L ? w ? ? ? L ? w = (0) (2. 94) is called the Lagrangian function, T is the kinetic energy, V is the potential energy, ? W is the nodal displacement and W is the nodal velocity. The kinetic energy of the element †e† can be expressed as Te = Aw2 2 ? ? W T W dx L (2. 96) ? and where ? is the density and W is the vector of velocities of element e. The expression for T using the eq(2. 9)to (2. 21) can be written as; ? ? N1 ? ? ? ? ? N 2? ? ? w1 ? 1 w2 ? 2 ? ? N 1 N 2 N 3 N 4 ? ? ? N 3? ? ? N4 ? ? w1 ? ? ? ? ? ?1 ? ? ? ? ? dx ? ? ? w2? ? ? ?2 Aw2 T = 2 e (2. 97) L 30 Rewriting the above expression we get; ? (N 1)2 ? ? ? N 2N 1 Aw2 ? Te = w1 ? 1 w2 ? 2 ? ? 2 L ? N 3N 1 ? N 4N 1 ? N 1N 2 N 1N 3 N 1N 4 w1 ? ? 2 (N 2) N 2N 3 N 2N 4? ? ? 1 ? ? ? ? ? dx ? N 3N 2 (N 3)2 N 3N 4? ?w2? ? 2 N 4N 2 N 4N 3 (N 4) ? 2 (2. 98) Recalling the shape functions derived in equations(2. 15) to (2. 18) N 1 = (1 ? x/l)2 (1 + 2x/l) N 2 = (1 ? x/l)2 x/l N 3 = (x/l)2 (3 ? 2x/l) N 4 = ? (1 ? x/l)(x/l)2 (2. 9) (2. 100) (2. 101) (2. 102) Substituting the shape functions from eqs(2. 99) to (2. 102) into eqs(2. 98) yields the elemental mass matrix for a pipe. ? ? 156 22l 54 ? 13l ? ? ? ? 2 2? ? 22l 4l 13l ? 3l ? Ml ? [M ]e = ? ? ? 420 ? 54 13l 156 ? 22l? ? ? ? 2 2 ? 13l ? 3l ? 22l 4l (2. 103) CHAPTER III FLOW INDUCED VIBRATIONS IN PIPES, A FINITE ELEMENT APPROACH 3. 1 Forming Global Sti? ness Matrix from Elemental Sti? ness Matrices Inorder to form a Global Matrix,we start with a 6Ãâ€"6 null matrix,with its six degrees of freedom being translation and rotation of each of the nodes. So our Global Sti? ness matrix looks like this: ? 0 ? ?0 ? ? ? ?0 =? ? ? 0 ? ? ? 0 ? ? 0 ? 0? ? 0? ? ? ? 0? ? ? 0? ? ? 0? ? ? 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 KGlobal (3. 1) 31 32 The two 4Ãâ€"4 element sti? ness matrices are: ? ? 12 6l ? 12 6l ? ? ? ? 4l2 ? 6l 2l2 ? EI ? 6l ? ? e [k1 ] = 3 ? ? l 12 ? 6l 12 ? 6l? ? ? ? ? 2 2 6l 2l ? 6l 4l ? 12 6l ? 12 6l ? (3. 2) ? ? ? ? 2 2? 4l ? 6l 2l ? EI ? 6l ? e [k2 ] = 3 ? ? l 12 ? 6l 12 ? 6l? ? ? ? ? 2 2 6l 2l ? 6l 4l (3. 3) We shall now build the global sti? ness matrix by inserting element 1 ? rst into the global sti? ness matrix. 6l ? 12 6l 0 0? ? 12 ? ? ? 6l 4l2 ? 6l 2l2 0 0? ? ? ? ? ? ? 12 ? 6l 12 ? l 0 0? EI ? ? = 3 ? ? l ? 6l 2 2 2l ? 6l 4l 0 0? ? ? ? ? ? 0 0 0 0 0 0? ? ? ? ? 0 0 0 0 0 0 ? ? KGlobal (3. 4) Inserting element 2 into the global sti? ness matrix ? ? 6l ? 12 6l 0 0 ? ? 12 ? ? ? 6l 4l2 ? 6l 2l2 0 0 ? ? ? ? ? ? ? EI 12 ? 6l (12 + 12) (? 6l + 6l) ? 12 6l ? ? KGlobal = 3 ? ? l ? 6l 2 2 2 2? ? 2l (? 6l + 6l) (4l + 4l ) ? 6l 2l ? ? ? ? ? 0 0 ? 12 ? 6l 12 ? 6l? ? ? ? ? 2 2 0 0 6l 2l ? 6l 4l (3. 5) 33 3. 2 Applying Boundary Conditions to Global Sti? ness Matrix for simply supported pipe with ? uid ? ow When the boundary conditions are applied to a simply supported pipe carrying ? uid, the 6Ãâ€"6 Global Sti? ess Matrix formulated in eq(3. 5) is mo di? ed to a 4Ãâ€"4 Global Sti? ness Matrix. It is as follows; Y 1 2 X L Figure 3. 1: Representation of Simply Supported Pipe Carrying Fluid ? ? 4l2 ?6l 2l2 0 KGlobalS ? ? ? ? EI 6l (12 + 12) (? 6l + 6l) 6l ? ? ? = 3 ? ? l ? 2l2 (? 6l + 6l) (4l2 + 4l2 ) 2l2 ? ? ? ? ? 2 2 0 6l 2l 4l (3. 6) Since the pipe is supported at the two ends the pipe does not de? ect causing its two translational degrees of freedom to go to zero. Hence we end up with the Sti? ness Matrix shown in eq(3. 6) 34 3. 3 Applying Boundary Conditions to Global Sti? ness Matrix for a cantilever pipe with ? id ? ow Y E, I 1 2 X L Figure 3. 2: Representation of Cantilever Pipe Carrying Fluid When the boundary conditions are applied to a Cantilever pipe carrying ? uid, the 6Ãâ€"6 Global Sti? ness Matrix formulated in eq(3. 5) is modi? ed to a 4Ãâ€"4 Global Sti? ness Matrix. It is as follows; ? (12 + 12) (? 6l + 6l) ? 12 6l ? KGlobalS ? ? ? ? ?(? 6l + 6l) (4l2 + 4l2 ) ? 6l 2l2 ? EI ? ? = 3 ? ? ? l ? ?12 ? 6l 12 ? 6l? ? ? ? 6l 2l2 ? 6l 4l2 (3. 7) Since the pipe is supported at one end the pipe does not de? ect or rotate at that end causing translational and rotational degrees of freedom at that end to go to zero.Hence we end up with the Sti? ness Matrix shown in eq(3. 8) 35 3. 4 MATLAB Programs for Assembling Global Matrices for Simply Supported and Cantilever pipe carrying ? uid In this section,we implement the method discussed in section(3. 1) to (3. 3) to form global matrices from the developed elemental matrices of a straight ? uid conveying pipe and these assembled matrices are later solved for the natural frequency and onset of instability of a cantlilever and simply supported pipe carrying ? uid utilizing MATLAB Programs. Consider a pipe of length L, modulus of elasticity E has ? uid ? wing with a velocity v through its inner cross-section having an outside diameter od,and thickness t1. The expression for critical velocity and natural frequency of the simply supported pipe carrying ? uid is given by; wn = ((3. 14)2 /L2 ) vc = (3. 14/L) (E ? I/M ) (3. 8) (3. 9) (E ? I/? A) 3. 5 MATLAB program for a simply supported pipe carrying ? uid The number of elements,density,length,modulus of elasticity of the pipe,density and velocity of ? uid ? owing through the pipe and the thickness of the pipe can be de? ned by the user. Refer to Appendix 1 for the complete MATLAB Program. 36 3. 6MATLAB program for a cantilever pipe carrying ? uid Figure 3. 3: Pinned-Free Pipe Carrying Fluid* The number of elements,density,length,modulus of elasticity of the pipe,density and velocity of ? uid ? owing through the pipe and the thickness of the pipe can be de? ned by the user. The expression for critical velocity and natural frequency of the cantilever pipe carrying ? uid is given by; wn = ((1. 875)2 /L2 ) (E ? I/M ) Where, wn = ((an2 )/L2 ) (EI/M )an = 1. 875, 4. 694, 7. 855 vc = (1. 875/L) (E ? I/? A) (3. 11) (3. 10) Refer to Appendix 2 for the complete MATLAB Program. 0 * Flow Induced Vibrat ions,Robert D.Blevins,Krieger. 1977,P 297 CHAPTER IV FLOW INDUCED VIBRATIONS IN PIPES, A FINITE ELEMENT APPROACH 4. 1 Parametric Study Parametric study has been carried out in this chapter. The study is carried out on a single span steel pipe with a 0. 01 m (0. 4 in. ) diameter and a . 0001 m (0. 004 in. ) thick wall. The other parameters are: Density of the pipe ? p (Kg/m3 ) 8000 Density of the ? uid ? f (Kg/m3 ) 1000 Length of the pipe L (m) 2 Number of elements n 10 Modulus Elasticity E (Gpa) 207 of MATLAB program for the simply supported pipe with ? uid ? ow is utilized for these set of parameters with varying ? uid velocity.Results from this study are shown in the form of graphs and tables. The fundamental frequency of vibration and the critical velocity of ? uid for a simply supported pipe 37 38 carrying ? uid are: ? n 21. 8582 rad/sec vc 16. 0553 m/sec Table 4. 1: Reduction of Fundamental Frequency for a Pinned-Pinned Pipe with increasing Flow Velocity Velocity of Fluid(v) Ve locity Ratio(v/vc) 0 2 4 6 8 10 12 14 16. 0553 0 0. 1246 0. 2491 0. 3737 0. 4983 0. 6228 0. 7474 0. 8720 1 Frequency(w) 21. 8806 21. 5619 20. 5830 18. 8644 16. 2206 12. 1602 3. 7349 0. 3935 0 Frequency Ratio(w/wn) 1 0. 9864 0. 9417 0. 8630 0. 7421 0. 5563 0. 709 0. 0180 0 39 Figure 4. 1: Reduction of Fundamental Frequency for a Pinned-Pinned Pipe with increasing Flow Velocity The fundamental frequency of vibration and the critical velocity of ? uid for a Cantilever pipe carrying ? uid are: ? n 7. 7940 rad/sec vc 9. 5872 m/sec 40 Figure 4. 2: Shape Function Plot for a Cantilever Pipe with increasing Flow Velocity Table 4. 2: Reduction of Fundamental Frequency for a Pinned-Free Pipe with increasing Flow Velocity Velocity of Fluid(v) Velocity Ratio(v/vc) 0 2 4 6 8 9 9. 5872 0 0. 2086 0. 4172 0. 6258 0. 8344 0. 9388 1 Frequency(w) 7. 7940 7. 5968 6. 9807 5. 8549 3. 825 1. 9897 0 Frequency Ratio(w/wn) 1 0. 9747 0. 8957 0. 7512 0. 4981 0. 2553 0 41 Figure 4. 3: Reduction of Fundamental Fr equency for a Cantilever Pipe with increasing Flow Velocity CHAPTER V FLOW INDUCED VIBRATIONS IN PIPES, A FINITE ELEMENT APPROACH E, I v L Figure 5. 1: Representation of Tapered Pipe Carrying Fluid 5. 1 Tapered Pipe Carrying Fluid Consider a pipe of length L, modulus of elasticity E. A ? uid ? ows through the pipe at a velocity v and density ? through the internal pipe cross-section. As the ? uid ? ows through the de? ecting pipe it is accelerated, because of the changing curvature 42 43 f the pipe and the lateral vibration of the pipeline. The vertical component of ? uid pressure applied to the ? uid element and the pressure force F per unit length applied on the ? uid element by the tube walls oppose these accelerations. The input parameters are given by the user. Density of the pipe ? p (Kg/m3 ) 8000 Density of the ? uid ? f (Kg/m3 ) 1000 Length of the pipe L (m) 2 Number of elements n 10 Modulus Elasticity E (Gpa) 207 of For these user de? ned values we introduce a taper in the pipe so that the material property and the length of the pipe with the taper or without the taper remain the same.This is done by keeping the inner diameter of the pipe constant and varying the outer diameter. Refer to ? gure (5. 2) The pipe tapers from one end having a thickness x to the other end having a thickness Pipe Carrying Fluid 9. 8mm OD= 10 mm L=2000 mm x mm t =0. 01 mm ID= 9. 8 mm Tapered Pipe Carrying Fluid Figure 5. 2: Introducing a Taper in the Pipe Carrying Fluid of t = 0. 01mm such that the volume of material is equal to the volume of material 44 for a pipe with no taper. The thickness x of the tapered pipe is now calculated: From ? gure(5. 2) we have †¢ Outer Diameter of the pipe with no taper(OD) 10 mm †¢ Inner Diameter of the pipe(ID) 9. mm †¢ Outer Diameter of thick end of the Tapered pipe (OD1 ) †¢ Length of the pipe(L) 2000 mm †¢ Thickness of thin end of the taper(t) 0. 01 mm †¢ Thickness of thick end of the taper x mm Volume of th e pipe without the taper: V1 = Volume of the pipe with the taper: ? ? L ? 2 V2 = [ (OD1 ) + (ID + 2t)2 ] ? [ (ID2 )] 4 4 3 4 (5. 2) ? (OD2 ? ID2 )L 4 (5. 1) Since the volume of material distributed over the length of the two pipes is equal We have, V1 = V2 (5. 3) Substituting the value for V1 and V2 from equations(5. 1) and (5. 2) into equation(5. 3) yields ? ? ? L ? 2 (OD2 ? ID2 )L = [ (OD1 ) + (ID + 2t)2 ] ? (ID2 )] 4 4 4 3 4 The outer diameter for the thick end of the tapered pipe can be expressed as (5. 4) OD1 = ID + 2x (5. 5) 45 Substituting values of outer diameter(OD),inner diameter(ID),length(L) and thickness(t) into equation (5. 6) yields ? 2 ? ? 2000 ? (10 ? 9. 82 )2000 = [ (9. 8 + 2x)2 + (9. 8 + 0. 02)2 ] ? [ (9. 82 )] 4 4 4 3 4 Solving equation (5. 6) yields (5. 6) x = 2. 24mm (5. 7) Substituting the value of thickness x into equation(5. 5) we get the outer diameter OD1 as OD1 = 14. 268mm (5. 8) Thus, the taper in the pipe varies from a outer diameters of 14. 268 mm to 9 . 82 mm. 46The following MATLAB program is utilized to calculate the fundamental natural frequency of vibration for a tapered pipe carrying ? uid. Refer to Appendix 3 for the complete MATLAB program. Results obtained from the program are given in table (5. 1) Table 5. 1: Reduction of Fundamental Frequency for a Tapered pipe with increasing Flow Velocity Velocity of Fluid(v) Velocity Ratio(v/vc) 0 20 40 60 80 100 103. 3487 0 0. 1935 0. 3870 0. 5806 0. 7741 0. 9676 1 Frequency(w) 40. 8228 40. 083 37. 7783 33. 5980 26. 5798 10. 7122 0 Frequency Ratio(w/wn) . 8100 0. 7784 0. 7337 0. 6525 0. 5162 0. 2080 0The fundamental frequency of vibration and the critical velocity of ? uid for a tapered pipe carrying ? uid obtained from the MATLAB program are: ? n 51. 4917 rad/sec vc 103. 3487 m/sec CHAPTER VI RESULTS AND DISCUSSIONS In the present work, we have utilized numerical method techniques to form the basic elemental matrices for the pinned-pinned and pinned-free pipe carrying ? uid. Matlab programs have been developed and utilized to form global matrices from these elemental matrices and fundamental frequency for free vibration has been calculated for various pipe con? gurations and varying ? uid ? ow velocities.Consider a pipe carrying ? uid having the following user de? ned parameters. E, I v L v Figure 6. 1: Representation of Pipe Carrying Fluid and Tapered Pipe Carrying Fluid 47 48 Density of the pipe ? p (Kg/m3 ) 8000 Density of the ? uid ? f (Kg/m3 ) 1000 Length of the pipe L (m) 2 Number of elements n 10 Modulus Elasticity E (Gpa) 207 of Refer to Appendix 1 and Appendix 3 for the complete MATLAB program Parametric study carried out on a pinned-pinned and tapered pipe for the same material of the pipe and subjected to the same conditions reveal that the tapered pipe is more stable than a pinned-pinned pipe.Comparing the following set of tables justi? es the above statement. The fundamental frequency of vibration and the critical velocity of ? uid for a tapered and a pinned-pinned pipe carrying ? uid are: ? nt 51. 4917 rad/sec ? np 21. 8582 rad/sec vct 103. 3487 m/sec vcp 16. 0553 m/sec Table 6. 1: Reduction of Fundamental Frequency for a Tapered Pipe with increasing Flow Velocity Velocity of Fluid(v) Velocity Ratio(v/vc) 0 20 40 60 80 100 103. 3487 0 0. 1935 0. 3870 0. 5806 0. 7741 0. 9676 1 Frequency(w) 40. 8228 40. 083 37. 7783 33. 5980 26. 5798 10. 7122 0 Frequency Ratio(w/wn) 0. 8100 0. 7784 0. 7337 0. 6525 0. 5162 0. 2080 0 9 Table 6. 2: Reduction of Fundamental Frequency for a Pinned-Pinned Pipe with increasing Flow Velocity Velocity of Fluid(v) Velocity Ratio(v/vc) 0 2 4 6 8 10 12 14 16. 0553 0 0. 1246 0. 2491 0. 3737 0. 4983 0. 6228 0. 7474 0. 8720 1 Frequency(w) 21. 8806 21. 5619 20. 5830 18. 8644 16. 2206 12. 1602 3. 7349 0. 3935 0 Frequency Ratio(w/wn) 1 0. 9864 0. 9417 0. 8630 0. 7421 0. 5563 0. 1709 0. 0180 0 The fundamental frequency for vibration and critical velocity for the onset of instability in tapered pipe is approxim ately three times larger than the pinned-pinned pipe,thus making it more stable. 50 6. 1 Contribution of the Thesis Developed Finite Element Model for vibration analysis of a Pipe Carrying Fluid. †¢ Implemented the above developed model to two di? erent pipe con? gurations: Simply Supported and Cantilever Pipe Carrying Fluid. †¢ Developed MATLAB Programs to solve the Finite Element Models. †¢ Determined the e? ect of ? uid velocities and density on the vibrations of a thin walled Simply Supported and Cantilever pipe carrying ? uid. †¢ The critical velocity and natural frequency of vibrations were determined for the above con? gurations. †¢ Study was carried out on a variable wall thickness pipe and the results obtained show that the critical ? id velocity can be increased when the wall thickness is tapered. 6. 2 Future Scope †¢ Turbulence in Two-Phase Fluids In single-phase ? ow,? uctuations are a direct consequence of turbulence developed in ? uid, whe reas the situation is clearly more complex in two-phase ? ow since the ? uctuation of the mixture itself is added to the inherent turbulence of each phase. †¢ Extend the study to a time dependent ? uid velocity ? owing through the pipe. BIBLIOGRAPHY [1] Doods. H. L and H. Runyan †E? ects of High-Velocity Fluid Flow in the Bending Vibrations and Static Divergence of a Simply Supported Pipe†.National Aeronautics and Space Administration Report NASA TN D-2870 June(1965). [2] Ashley,H and G. Haviland †Bending Vibrations of a Pipe Line Containing Flowing Fluid†. J. Appl. Mech. 17,229-232(1950). [3] Housner,G. W †Bending Vibrations of a Pipe Line Containing Flowing Fluid†. J. Appl. Mech. 19,205-208(1952). [4] Long. R. H †Experimental and Theoretical Study of Transverse Vibration of a tube Containing Flowing Fluid†. J. Appl. Mech. 22,65-68(1955). [5] Liu. H. S and C. D. Mote †Dynamic Response of Pipes Transporting Fluids†. J. Eng. for Industry 96,591-596(1974). 6] Niordson,F. I. N †Vibrations of a Cylinderical Tube Containing Flowing Fluid†. Trans. Roy. Inst. Technol. Stockholm 73(1953). [7] Handelman,G. H †A Note on the transverse Vibration of a tube Containing Flowing Fluid†. Quarterly of Applied Mathematics 13,326-329(1955). [8] Nemat-Nassar,S. S. N. Prasad and G. Herrmann †Destabilizing E? ect on VelocityDependent Forces in Nonconservative Systems†. AIAA J. 4,1276-1280(1966). 51 52 [9] Naguleswaran,S and C. J. H. Williams †Lateral Vibrations of a Pipe Conveying a Fluid†. J. Mech. Eng. Sci. 10,228-238(1968). [10] Herrmann. G and R. W.Bungay †On the Stability of Elastic Systems Subjected to Nonconservative Forces†. J. Appl. Mech. 31,435-440(1964). [11] Gregory. R. W and M. P. Paidoussis †Unstable Oscillations of Tubular Cantilevers Conveying Fluid-I Theory†. Proc. Roy. Soc. (London). Ser. A 293,512-527(1966). [12] S. S. Rao †The Finite Element Method in Engineering†. Pergamon Press Inc. 245294(1982). [13] Michael. R. Hatch †Vibration Simulation Using Matlab and Ansys†. Chapman and Hall/CRC 349-361,392(2001). [14] Robert D. Blevins †Flow Induced Vibrations†. Krieger 289,297(1977). Appendices 53 54 0. 1 MATLAB program for Simply Supported Pipe Carrying FluidMATLAB program for Simply Supported Pipe Carrying Fluid. % The f o l l o w i n g MATLAB Program c a l c u l a t e s t h e Fundamental % N a t u r a l f r e q u e n c y o f v i b r a t i o n , f r e q u e n c y r a t i o (w/wn) % and v e l o c i t y r a t i o ( v / vc ) , f o r a % simply supported pipe carrying f l u i d . % I n o r d e r t o perform t h e above t a s k t h e program a s s e m b l e s % E l e m e n t a l S t i f f n e s s , D i s s i p a t i o n , and I n e r t i a m a t r i c e s % t o form G l o b a l M a t r i c e s which are used t o c a l c u l a t e % Fundamental N a t u r a l % Frequency w . lc ; n um elements =input ( ’ Input number o f e l e m e n t s f o r beam : ’ ) ; % num elements = The u s e r e n t e r s t h e number o f e l e m e n t s % i n which t h e p i p e % has t o be d i v i d e d . n=1: num elements +1;% Number o f nodes ( n ) i s e q u a l t o number o f %e l e m e n t s p l u s one n o d e l =1: num elements ; node2 =2: num elements +1; max nodel=max( n o d e l ) ; max node2=max( node2 ) ; max node used=max( [ max nodel max node2 ] ) ; mnu=max node used ; k=zeros (2? mnu ) ;% C r e a t i n g a G l o b a l S t i f f n e s s Matrix o f z e r o s 55 m =zeros (2? nu ) ;% C r e a t i n g G l o b a l Mass Matrix o f z e r o s x=zeros (2? mnu ) ;% C r e a t i n g G l o b a l Matrix o f z e r o s % f o r t h e f o r c e t h a t conforms f l u i d % to the curvature of the % pipe d=zeros (2? mnu ) ;% C r e a t i n g G l o b a l D i s s i p a t i o n Matrix o f z e r o s %( C o r i o l i s Component ) t=num elements ? 2 ; L=2; % T o t a l l e n g t h o f t h e p i p e i n meters l=L/ num elements ; % Length o f an e l e m e n t t1 =. 0001; od = . 0 1 ; i d=od? 2? t 1 % t h i c k n e s s o f t h e p i p e i n meter % outer diameter of the pipe % inner diameter of the pipeI=pi ? ( od? 4? i d ? 4)/64 % moment o f i n e r t i a o f t h e p i p e E=207? 10? 9; roh =8000; rohw =1000; % Modulus o f e l a s t i c i t y o f t h e p i p e % Density of the pipe % d e n s i t y o f water ( FLuid ) M =roh ? pi ? ( od? 2? i d ? 2)/4 + rohw? pi ? . 2 5 ? i d ? 2 ; % mass per u n i t l e n g t h o f % the pipe + f l u i d rohA=rohw? pi ? ( . 2 5 ? i d ? 2 ) ; l=L/ num elements ; v=0 % v e l o c i t y o f t h e f l u i d f l o w i n g t h r o u g h t h e p i p e %v =16. 0553 z=rohA/M i=sqrt ( ? 1); wn= ( ( 3 . 1 4 ) ? 2 /L? 2)? sqrt (E? I /M) % N a t u r a l Frequency vc =(3. 14/L)? sqrt (E?I /rohA ) % C r i t i c a l V e l o c i t y 56 % Assembling G l o b a l S t i f f n e s s , D i s s i p a t i o n and I n e r t i a M a t r i c e s for j =1: nu m elements d o f 1 =2? n o d e l ( j ) ? 1; d o f 2 =2? n o d e l ( j ) ; d o f 3 =2? node2 ( j ) ? 1; d o f 4 =2? node2 ( j ) ; % S t i f f n e s s Matrix Assembly k ( dof1 , d o f 1 )=k ( dof1 , d o f 1 )+ (12? E? I / l ? 3 ) ; k ( dof2 , d o f 1 )=k ( dof2 , d o f 1 )+ (6? E? I / l ? 2 ) ; k ( dof3 , d o f 1 )=k ( dof3 , d o f 1 )+ (? 12? E? I / l ? 3 ) ; k ( dof4 , d o f 1 )=k ( dof4 , d o f 1 )+ (6? E? I / l ? 2 ) ; k ( dof1 , d o f 2 )=k ( dof1 , d o f 2 )+ (6? E?I / l ? 2 ) ; k ( dof2 , d o f 2 )=k ( dof2 , d o f 2 )+ (4? E? I / l ) ; k ( dof3 , d o f 2 )=k ( dof3 , d o f 2 )+ (? 6? E? I / l ? 2 ) ; k ( dof4 , d o f 2 )=k ( dof4 , d o f 2 )+ (2? E? I / l ) ; k ( dof1 , d o f 3 )=k ( dof1 , d o f 3 )+ (? 12? E? I / l ? 3 ) ; k ( dof2 , d o f 3 )=k ( dof2 , d o f 3 )+ (? 6? E? I / l ? 2 ) ; k ( dof3 , d o f 3 )=k ( dof3 , d o f 3 )+ (12? E? I / l ? 3 ) ; k ( dof4 , d o f 3 )=k ( dof4 , d o f 3 )+ (? 6? E? I / l ? 2 ) ; k ( dof1 , d o f 4 )=k ( dof1 , d o f 4 )+ (6? E? I / l ? 2 ) ; k ( dof2 , d o f 4 )=k ( dof2 , d o f 4 )+ (2? E? I / l ) ; k ( dof3 , d o f 4 )=k ( dof3 , d o f 4 )+ (? ? E? I / l ? 2 ) ; k ( dof4 , d o f 4 )=k ( dof4 , d o f 4 )+ (4? E? I / l ) ; % 57 % Matrix a s s e m b l y f o r t h e second term i e % f o r t h e f o r c e t h a t conforms % f l u i d to the curvature of the pipe x ( dof1 , d o f 1 )=x ( dof1 , d o f 1 )+ ( ( 3 6 ? rohA? v ? 2)/30? l ) ; x ( dof2 , d o f 1 )=x ( dof2 , d o f 1 )+ ( ( 3 ? rohA? v ? 2)/30? l ) ; x ( dof3 , d o f 1 )=x ( dof3 , d o f 1 )+ (( ? 36? rohA? v ? 2)/30? l ) ; x ( dof4 , d o f 1 )=x ( dof4 , d o f 1 )+ ( ( 3 ? rohA? v ? 2)/30? l ) ; x ( dof1 , d o f 2 )=x ( dof1 , d o f 2 )+ ( ( 3 ? ohA? v ? 2)/30? l ) ; x ( dof2 , d o f 2 )=x ( dof2 , d o f 2 )+ ( ( 4 ? rohA? v ? 2)/30? l ) ; x ( dof3 , d o f 2 )=x ( dof3 , d o f 2 )+ (( ? 3? rohA? v ? 2)/30? l ) ; x ( dof4 , d o f 2 )=x ( dof4 , d o f 2 )+ (( ? 1? rohA? v ? 2)/30? l ) ; x ( dof1 , d o f 3 )=x ( dof1 , d o f 3 )+ (( ? 36? rohA? v ? 2)/30? l ) ; x ( dof2 , d o f 3 )=x ( dof2 , d o f 3 )+ (( ? 3? rohA? v ? 2)/30? l ) ; x ( dof3 , d o f 3 )=x ( dof3 , d o f 3 )+ ( ( 3 6 ? rohA? v ? 2)/30? l ) ; x ( dof4 , d o f 3 )=x ( dof4 , d o f 3 )+ (( ? 3? rohA? v ? 2)/30? l ) ; x ( dof1 , d o f 4 )=x ( dof1 , d o f 4 )+ ( ( 3 ? rohA? v ? 2)/30? ) ; x ( dof2 , d o f 4 )=x ( dof2 , d o f 4 )+ (( ? 1? rohA? v ? 2)/30? l ) ; x ( dof3 , d o f 4 )=x ( dof3 , d o f 4 )+ (( ? 3? rohA? v ? 2)/30? l ) ; x ( dof4 , d o f 4 )=x ( dof4 , d o f 4 )+ ( ( 4 ? rohA? v ? 2)/30? l ) ; % % D i s s i p a t i o n Matrix Assembly d ( dof1 , d o f 1 )=d ( dof1 , d o f 1 )+ (2? ( ? 30? rohA? v ) / 6 0 ) ; d ( dof2 , d o f 1 )=d ( dof2 , d o f 1 )+ ( 2 ? ( 6 ? rohA? v ) / 6 0 ) ; d ( dof3 , d o f 1 )=d ( dof3 , d o f 1 )+ ( 2 ? ( 3 0 ? rohA? v ) / 6 0 ) ; 58 d ( dof4 , d o f 1 )=d ( dof4 , d o f 1 )+ (2? ( ? 6? rohA? ) / 6 0 ) ; d ( dof1 , d o f 2 )=d ( dof1 , d o f 2 )+ (2? ( ? 6? rohA? v ) / 6 0 ) ; d ( dof2 , d o f 2 )=d ( dof2 , d o f 2 )+ ( 2 ? ( 0 ? ro hA? v ) / 6 0 ) ; d ( dof3 , d o f 2 )=d ( dof3 , d o f 2 )+ ( 2 ? ( 6 ? rohA? v ) / 6 0 ) ; d ( dof4 , d o f 2 )=d ( dof4 , d o f 2 )+ (2? ( ? 1? rohA? v ) / 6 0 ) ; d ( dof1 , d o f 3 )=d ( dof1 , d o f 3 )+ (2? ( ? 30? rohA? v ) / 6 0 ) ; d ( dof2 , d o f 3 )=d ( dof2 , d o f 3 )+ (2? ( ? 6? rohA? v ) / 6 0 ) ; d ( dof3 , d o f 3 )=d ( dof3 , d o f 3 )+ ( 2 ? ( 3 0 ? rohA? v ) / 6 0 ) ; d ( dof4 , d o f 3 )=d ( dof4 , d o f 3 )+ ( 2 ? ( 6 ? rohA? v ) / 6 0 ) ; ( dof1 , d o f 4 )=d ( dof1 , d o f 4 )+ ( 2 ? ( 6 ? rohA? v ) / 6 0 ) ; d ( dof2 , d o f 4 )=d ( dof2 , d o f 4 )+ ( 2 ? ( 1 ? rohA? v ) / 6 0 ) ; d ( dof3 , d o f 4 )=d ( dof3 , d o f 4 )+ (2? ( ? 6? rohA? v ) / 6 0 ) ; d ( dof4 , d o f 4 )=d ( dof4 , d o f 4 )+ ( 2 ? ( 0 ? rohA? v ) / 6 0 ) ; % % I n e r t i a Matrix Assembly m( dof1 , d o f 1 )=m( dof1 , d o f 1 )+ (156? M? l / 4 2 0 ) ; m( dof2 , d o f 1 )=m( dof2 , d o f 1 )+ (22? l ? 2? M/ 4 2 0 ) ; m( dof3 , d o f 1 )=m( dof3 , d o f 1 )+ (54? l ? M/ 4 2 0 ) ; m( d of4 , d o f 1 )=m( dof4 , d o f 1 )+ (? 3? l ? 2? M/ 4 2 0 ) ; m( dof1 , d o f 2 )=m( dof1 , d o f 2 )+ (22? l ? 2? M/ 4 2 0 ) ; m( dof2 , d o f 2 )=m( dof2 , d o f 2 )+ (4? M? l ? 3 / 4 2 0 ) ; m( dof3 , d o f 2 )=m( dof3 , d o f 2 )+ (13? l ? 2? M/ 4 2 0 ) ; m( dof4 , d o f 2 )=m( dof4 , d o f 2 )+ (? 3? M? l ? 3 / 4 2 0 ) ; 59 m( dof1 , d o f 3 )=m( dof1 , d o f 3 )+ (54? M? l / 4 2 0 ) ; m( dof2 , d o f 3 )=m( dof2 , d o f 3 )+ (13? l ? 2? M/ 4 2 0 ) ; m( dof3 , d o f 3 )=m( dof3 , d o f 3 )+ (156? l ? M/ 4 2 0 ) ; m( dof4 , d o f 3 )=m( dof4 , d o f 3 )+ (? 22? l ? 2? M/ 4 2 0 ) ; m( dof1 , d o f 4 )=m( dof1 , d o f 4 )+ (? 13? l ? 2?M/ 4 2 0 ) ; m( dof2 , d o f 4 )=m( dof2 , d o f 4 )+ (? 3? M? l ? 3 / 4 2 0 ) ; m( dof3 , d o f 4 )=m( dof3 , d o f 4 )+ (? 22? l ? 2? M/ 4 2 0 ) ; m( dof4 , d o f 4 )=m( dof4 , d o f 4 )+ (4? M? l ? 3 / 4 2 0 ) ; end k ( 1 : 1 , : ) = [ ] ;% A p p l y i n g Boundary c o n d i t i o n s k(: ,1:1)=[]; k ( ( 2 ? mnu? 2 ) : ( 2 ? mnu? 2 ) , : ) = [ ] ; k ( : , ( 2 ? mnu? 2 ) : ( 2 ? mnu? 2 ) ) = [ ] ; k x(1:1 ,:)=[]; x(: ,1:1)=[]; x ( ( 2 ? mnu? 2 ) : ( 2 ? mnu? 2 ) , : ) = [ ] ; x ( : , ( 2 ? mnu? 2 ) : ( 2 ? mnu? 2 ) ) = [ ] ; x; % G l o b a l Matrix f o r t h e % Force t h a t conforms f l u i d t o p i p e x1=? d(1:1 ,:)=[]; d(: ,1:1)=[]; d ( ( 2 ? mnu? 2 ) : ( 2 ? mnu? 2 ) , : ) = [ ] ; % G l o b a l S t i f f n e s s Matrix 60 d ( : , ( 2 ? mnu? 2 ) : ( 2 ? mnu? 2 ) ) = [ ] ; d d1=(? d ) Kg lobal=k+10? x1 ; m( 1 : 1 , : ) = [ ] ; m( : , 1 : 1 ) = [ ] ; m( ( 2 ? mnu? 2 ) : ( 2 ? mnu? 2 ) , : ) = [ ] ; m( : , ( 2 ? mnu? 2 ) : ( 2 ? mnu? 2 ) ) = [ ] ; m; eye ( t ) ; zeros ( t ) ; H=[? inv (m) ? ( d1 ) ? inv (m)? Kglobal ; eye ( t ) zeros ( t ) ] ; Evalue=eig (H) % E i g e n v a l u e s v r a t i o=v/ vc % V e l o c i t y Ratio % G l o b a l Mass Matrix % G l o b a l D i s s i p a t i o nMatrix i v 2=imag ( Evalue ) ; i v 2 1=min( abs ( i v 2 ) ) ; w1 = ( i v 2 1 ) wn w r a t i o=w1/wn vc % Frequency Ratio % Fundamental N a t u r a l f r e q u e n c y 61 0. 2 MATLAB Program for Cantilever Pipe Carrying Fluid MATLAB Program for Cantilever Pipe Carrying Fluid. % The f o l l o w i n g MATLAB Program c a l c u l a t e s t h e Fundamental % N a t u r a l f r e q u e n c y o f v i b r a t i o n , f r e q u e n c y r a t i o (w/wn) % and v e l o c i t y r a t i o ( v / vc ) , f o r a c a n t i l e v e r p i p e % carrying f l u i d . I n o r d e r t o perform t h e above t a s k t h e program a s s e m b l e s % E l e m e n t a l S t i f f n e s s , D i s s i p a t i o n , and I n e r t i a m a t r i c e s % t o form G l o b a l M a t r i c e s which are used % t o c a l c u l a t e Fundamental N a t u r a l % Frequency w . clc ; num elements =input ( ’ Input number o f e l e m e n t s f o r Pipe : ’ ) ; % num elements = The u s e r e n t e r s t h e number o f e l e m e n t s % i n which t h e p i p e has t o be d i v i d e d . =1: num elements +1;% Number o f nodes ( n ) i s % e q u a l t o num ber o f e l e m e n t s p l u s one n o d e l =1: num elements ; % Parameters used i n t h e l o o p s node2 =2: num elements +1; max nodel=max( n o d e l ) ; max node2=max( node2 ) ; max node used=max( [ max nodel max node2 ] ) ; mnu=max node used ; k=zeros (2? mnu ) ;% C r e a t i n g a G l o b a l S t i f f n e s s Matrix o f z e r o s 62 m =zeros (2? mnu ) ;% C r e a t i n g G l o b a l Mass Matrix o f z e r o s